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a/29AzT4oBgHgl3EQfuf0F/content/tmp_files/2301.01690v1.pdf.txt b/29AzT4oBgHgl3EQfuf0F/content/tmp_files/2301.01690v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b17e3d6f78680081370f987e8b22bcf33f922f9f --- /dev/null +++ b/29AzT4oBgHgl3EQfuf0F/content/tmp_files/2301.01690v1.pdf.txt @@ -0,0 +1,1345 @@ +arXiv:2301.01690v1 [cs.LO] 4 Jan 2023 +Proofs as stateful programs: A first-order logic with abstract Hoare +triples, and an interpretation into an imperative language +Thomas Powell +January 5, 2023 +Abstract +We introduce an extension of first-order logic that comes equipped with additional predicates for +reasoning about an abstract state. Sequents in the logic comprise a main formula together with pre- +and postconditions in the style of Hoare logic, and the axioms and rules of the logic ensure that the +assertions about the state compose in the correct way. The main result of the paper is a realizability +interpretation of our logic that extracts programs into a mixed functional/imperative language. +All +programs expressible in this language act on the state in a sequential manner, and we make this intuition +precise by interpreting them in a semantic metatheory using the state monad. Our basic framework is +very general, and our intention is that it can be instantiated and extended in a variety of different ways. +We outline in detail one such extension: A monadic version of Heyting arithmetic with a wellfounded +while rule, and conclude by outlining several other directions for future work. +1 +Introduction +The Curry-Howard correspondence lies at the heart of theoretical computer science. +Over the years, a +multitude of different techniques for extracting programs from proofs have been developed, the majority +of which translate formal proof systems into lambda calculi. As such, programs extracted from proofs are +typically conceived as pure functional programs. +Everyday programmers, on the other hand, often think and write in an imperative paradigm, in terms of +instructions that change some underlying global state. This is reinforced by the fact that many of the most +popular programming languages, including C and Python, lean towards this style. Imperative programs are +nevertheless highly complex from a mathematical perspective, and while systems such as Hoare logic [15] or +separation logic [27] have been designed to reason about them, the formal extraction of imperative programs +from proofs has received comparatively little attention. +In this paper, we propose a new idea in this direction, developing a formal system SL that enriches +ordinary first-order logic with Hoare triples for reasoning about an abstract global state. Sequents will have +the form Γ ⊢ {α · A · β}, where A is a formula and α, β assertions about the state, and proofs in the logic +will include both ordinary introduction and elimination rules for predicate logic, together with special rules +for reasoning about the state. We then construct a stateful realizability interpretation (based on Kreisel’s +modified realizability [17]) that relates formulas in SL to terms in a mixed functional/imperative language +ST. Our main result is a soundness theorem, which confirms that whenever a formula is provable in SL, we +can extract a corresponding stateful realizing term in ST. While our initial soundness theorem focuses on +pure predicate logic, we subsequently show that it can be extended to arithmetic, where in particular we are +then able to extract programs that contain both recursion and controlled while loops. +We are not the first to adapt traditional methods to extract imperative programs: A major achievement +in this direction, for example, is the monograph [22], which sets up a variant of intuitionistic Hoare logic +alongside a realizability translation into a standard imperative language. Other relevant examples include +[3, 8, 10, 13, 19, 32]. However, these and almost all other prior work in this direction tend to focus on +formal verification, with an eye towards using proof interpretations as a method for the synthesis of correct- +by-construction software. In concrete terms, this means that the formal systems tend to be quite detailed +1 + +and oriented towards program analysis, while the starting point is typically a program for which we want to +construct a verification proof, rather than a proof from which we hope to extract a potentially unfamiliar +program. +Our approach, on the other hand, is much more abstract, with an emphasis on potential applications in +logic and proof theory. Our basic system SL makes almost no assumptions about the structure of the state +and what we are allowed to do with it. Rather, we focus on producing a general framework for reasoning +about ‘stateful formulas’, which can then be instantiated with additional axioms to model concrete scenarios. +The simplicity and generality of our framework is its most important feature, and we consider this work to +be a first step towards a number of potentially interesting applications. For this reason, we include not only +an extension of our system to a monadic theory of arithmetic, but conclude by sketching out some additional +ways in which we conjecture that our logic and interpretation could be used and expanded, including the +computational semantics of proofs and probabilistic logic. +We take ideas from three main sources. +The first is a case study of Berger et al. +[6], in which a +realizability interpretation is used to extract a version of in-place quicksort, and where the imperative nature +of the extracted program is presented in a semantic way using the state monad. +While their program +behaves imperatively “by-chance”, terms extracted from our logic are forced to be imperative, and thus +our framework offers one potential solution to their open problem of designing a proof calculus which only +yields imperative programs. Indeed, an implementation of the insert sort algorithm is formally extracted +in Section 6 below. Our second source of inspiration is the thesis of Birolo [7], where a general monadic +realizability interpretation is defined and then used to give an alternative, semantic presentation of learning- +based interactive realizability [2, 4]. However, our work goes beyond this in that it also involves a monadic +extension of the target logic, whereas Birolo’s applies to standard first-order logic. Finally, a number of +ideas are taken from the author’s previous work [24] on extracting stateful programs using the Dialectica +interpretation. While there the state is used in a very specific and restricted way, unlike our more general +presentation here, we use an analogous call-by-value monadic translation on terms. +It is important to stress that we do not claim that our work represents an optimal or complete method +for extracting imperative programs from proofs, nor do we claim that it is superior to alternative methods, +including the aforementioned works in the direction of verification, or, for instance, techniques based on +Krivine’s classical realizability [18], which could be viewed as imperative in nature. We simply offer what +we consider to be a new and interesting perspective that emphasises abstraction and simplicity, and propose +that our framework could prove valuable in a number of different contexts. +Overview of the paper +The main technical work that follows involves the design of three different systems, a realizability interpre- +tation that connects them, and an instantiation of this framework in the setting of first-order arithmetic, +namely: +• A novel extension SL of predicate logic with abstract Hoare triples, which can be extended with +additional axioms for characterising the state (Section 2). +• A standard calculus ST for lambda terms with imperative commands, which can again be extended +with additional constants for interacting with the state (Section 3). +• A metalanguage Sω into which both SL and ST can be embedded (Section 4), which is used to formulate +the realizability relation and prove its soundness (Section 5). +• An instantiation of SL as a theory of arithmetic, with programs extracted into an extension of ST with +recursion and while loops (Section 6). +Concrete examples are given, and potential applications surveyed in Section 7. +2 + +2 +The system SL: First-order logic with state +We begin by introducing our target theory SL from which stateful programs will be extracted. This is an +extension of ordinary first-order logic in the sense that the latter can always be embedded into SL (we will +make this precise in Proposition 2.1 below). Ultimately, we are interested not so much in SL on its own, but +in theories of the form SL + ∆H + ∆S, where ∆H and ∆S are collections of (respectively non-computational +and computational) axioms that together characterise the state. Several concrete examples will be given to +illustrate this, and in Section 6 we present a variant of SL that represents a theory of first-order arithmetic +with state. +2.1 +Intuitionistic first-order logic +Before defining SL, we give a standard presentation of first-order intuitionistic predicate logic PL, which +serves as an opportunity to fix our basic style of formal reasoning. The language of PL consists of the +logical constants ∧, ∨, ⇒, ∀, ∃, ⊤, ⊥, variables x, y, z, . . ., along with function symbols f, g, h, . . . and predicate +symbols P, Q, R, . . ., each with a fixed arity. We assume the existence of at least one constant c. Terms are +built from variables and function symbols as usual, and formulas are built from prime formulas P(t1, . . . , tn), +⊤ and ⊥ using the logical constants. We use the usual abbreviation ¬A :≡ A ⇒ ⊥. We work in a sequent +style natural deduction calculus, where sequents have the form Γ ⊢I A for some context Γ and formula A, +and a context is a set of labelled assumptions of the form Au1 +1 , . . . , Aun +n +for pairwise distinct labels ui. The +axioms and rules of PL are as in Figure 1. +Figure 1: Axioms and rules of PL +Propositional logic +Γ ⊢I A +if Au ∈ Γ for some u +Γ ⊢I ⊤ +Γ ⊢I A +Γ ⊢I B +∧I +Γ ⊢I A ∧ B +Γ ⊢I A ∧ B +∧EL +Γ ⊢I A +Γ ⊢I A ∧ B +∧ER +Γ ⊢I B +Γ ⊢I A +∨IL +Γ ⊢I A ∨ B +Γ ⊢I B +∨IR +Γ ⊢I A ∨ B +Γ ⊢I A ∨ B +Γ, Au ⊢I C +Γ, Bv ⊢I C +∨E +Γ ⊢I C +Γ, Au ⊢I B +⇒I +Γ ⊢I A ⇒ B +Γ ⊢I A ⇒ B +Γ ⊢I A +⇒E +Γ ⊢I B +Γ ⊢I ⊥ +⊥E +Γ ⊢I A +Quantifier rules +Γ ⊢I A[y/x] +∀I +Γ ⊢I ∀xA +Γ ⊢I ∀xA +∀E +Γ ⊢I A[t/x] +Γ ⊢I A[t/x] +∃I +Γ ⊢I ∃xA +Γ ⊢I ∃xA +Γ, A[y/x]u ⊢I C +∃E +Γ ⊢I C +for ∀I, y ≡ x or y not free in A, and y not free in Γ +for ∃E, y ≡ x or y not free in A, and y not free in C or Γ. +3 + +2.2 +Stateful first-order logic +We now define our new logical system SL, which is an extension of ordinary first-order logic with new state +propositions. To be more precise, we extend the language of PL with a ternary operation {−·−·−}, together +with special state predicate symbols p, q, r, . . ., which also have a fixed arity. Terms of SL are the same as +those of PL. On the other hand, there are two kinds of formulas in SL: state formulas and main formulas. +A state formula is defined using state predicate symbols and propositional connectives as follows: +• ⊤ and ⊥ are state formulas, +• if p a state predicate symbol of arity n and t1, . . . , tn are terms, then p(t1, . . . , tn) is a state formula, +• if α, β are state formulas, so are α ∧ β, α ∨ β, α ⇒ β. +A main formula (or just formula) of SL is now defined as: +• ⊤ and ⊥ are formulas, +• if P is an ordinary predicate symbol of arity n and t1, . . . , tn are terms, then P(t1, . . . , tn) is a formula, +• if A, B are formulas, so are A ∧ B, A ∨ B and ∃xA, +• if A, B are formulas and α, β state formulas, then A ⇒ {α · B · β} and ∀x {α · A · β} are formulas. +The notions of free and bound variables, along with substitution α[t/x] and A[t/x] can be easily defined for +both state and main formulas. +Analogous to the construction of formulas, our basic proof system uses the auxiliary notion of a state +proof in order to define a main proof. A state sequent has the form Γ ⊢H α where α is a state formula and +Γ a set of labelled state formulas. A proof of Γ ⊢H α in SL is built from the axioms and rules of classical +propositional logic i.e. the propositional axioms and rules as set out in Figure 1 plus the law of excluded +middle Γ ⊢H α ∨ ¬α, together with a set ∆H of as yet unspecified state axioms of the form Γ ⊢H α. +A main sequent of SL has the form Γ ⊢S {α · A · β}, where A is a formula and α, β state formulas, and +Γ is a set of labelled main formulas. A proof of Γ ⊢S {α · A · β} in SL uses the axioms and rules given in +Figure 2, together with a set ∆S of additional axioms. +We now make precise what we mean when we characterise SL as an extension of standard first-order +logic. The following is provable with an easy induction over derivations in PL: +Proposition 2.1. For any formula A of PL and state formula α, define the main formula Aα of SL by +• Qα := Q for Q atomic, +• (A ∧ B)α := Aα ∧ Bα, (A ∨ B)α := Aα ∨ Bα and (∃x A)α := ∃x Aα, +• (A ⇒ B)α := Aα ⇒ {α · Bα · α} and (∀x A)α := ∀x {α · Aα · α}. +Then whenever Γ ⊢I A is provable in PL, we have that Γα, ∆ ⊢S {α · Aα · α} is provable in SL, where ∆ is +arbitrary and Γα := (A1)u1 +α , . . . , (An)un +α +for Γ := Au1 +1 , . . . , Aun +n . +2.3 +The intuition behind SL +The intended semantic meaning of Γ ⊢H α is that α can be inferred from the assumptions Γ for any fixed state. +More specifically, if we imagine a semantic variant [α](π) of each state formula where now the dependency +on an underlying state π is made explicit, the semantics of Γ ⊢H α is just +[Γ](π) ⇒ [α](π) +4 + +Figure 2: Axioms and rules of SL +Propositional axioms and rules +Γ ⊢S {α · A · α} +if Au ∈ Γ for some u +Γ ⊢S {α · ⊤ · α} +Γ ⊢S {α · A · β} +Γ ⊢S {β · B · γ} +∧SI +Γ ⊢S {α · A ∧ B · γ} +Γ ⊢S {α · A ∧ B · β} +∧SEL +Γ ⊢S {α · A · β} +Γ ⊢S {α · A ∧ B · β} +∧SER +Γ ⊢S {α · B · β} +Γ ⊢S {α · A · β} +∨SIL +Γ ⊢S {α · A ∨ B · β} +Γ ⊢S {α · B · β} +∨SIR +Γ ⊢S {α · A ∨ B · β} +Γ ⊢S {α · A ∨ B · β} +Γ, Au ⊢S {β · C · γ} +Γ, Bv ⊢S {β · C · γ} +∨SE +Γ ⊢S {α · C · γ} +Γ, Au ⊢S {α · B · β} +⇒SI +Γ ⊢S {γ · A ⇒ {α · B · β} · γ} +Γ ⊢S {α · A ⇒ {γ · B · δ} · β} +Γ ⊢S {β · A · γ} +⇒SE +Γ ⊢S {α · B · δ} +Γ ⊢S {α · ⊥ · β} +⊥SE +Γ ⊢S {α · A · γ} +Quantifier rules +Γ ⊢S {α[y/x] · A[y/x] · β[y/x]} +∀SI +Γ ⊢S {γ · ∀x {α · A · β} · γ} +Γ ⊢S {α · ∀x {β · A · γ} · β[t/x]} +∀SE +Γ ⊢S {α · A[t/x] · γ[t/x]} +Γ ⊢S {α · A[t/x] · β} +∃SI +Γ ⊢S {α · ∃xA · β} +Γ ⊢S {α · ∃xA · β} +Γ, A[y/x]u ⊢S {β · C · γ} +∃SE +Γ ⊢S {α · C · γ} +for ∀SI, y ≡ x or y not free in A, α, β, and y not free in Γ +for ∃SE, y ≡ x or y not free in A, and y not free in C, α, β, γ or Γ. +Basic Hoare rules +α ⊢H β +Γ ⊢S {β · A · γ} +γ ⊢H δ +cons +Γ ⊢S {α · A · δ} +⊢H α ∨ β +Γ ⊢S {α ∧ γ · A · δ} +Γ ⊢S {β ∧ γ · A · δ} +cond +Γ ⊢S {γ · A · δ} +Additional axioms +state axioms ∆H of the form Γ ⊢H α +main axioms ∆S of the form Γ ⊢S {α · A · β} +5 + +On the other hand, the intended meaning of Γ ⊢S {α · A · β} is that from assumptions Γ, if α holds with +respect to some initial state, then we can infer that A is true and β holds with respect to some modified +state, or more precisely: +[Γ] ⇒ (∃π [α](π) ⇒ ([A] ∧ ∃π′ [β](π′))) +(1) +In particular, the computational interpretation of (1) above will be a program that takes some input state +π satisfying [α](π) and returns a realizer-state pair ⟨x, π′⟩ such that x realizes A and [β](π′) holds. +Our semantic interpretation [·] will be properly defined in Section 4. Crucially, in SL the state is implicit, +and so there are no variables or terms of state type. The state will rather be made explicit in our metatheory +Sω. The main axioms and rules of SL simply describe how this semantic interpretation propagates in a +call-by-value manner through the usual axioms and rules of first-order logic. The state itself is brought into +play through the Hoare rules along with the additional axioms ∆H and ∆S. +The two Hoare rules of SL correspond to the consequence and conditional rules of traditional Hoare +logic. The usual conditional rule falls out as a special case of ours since we assume Γ ⊢H α ∨ ¬α. Some of +the other traditional Hoare rules are derivable: The empty statement axiom corresponds to our own axiom +Γ ⊢S {α · ⊤ · α} while composition can be viewed as the special case of ∧SI for A = B = ⊤. In Section 6 we +add a controlled while loop to our logic. But for now, we illustrate our logic with some very straightforward +scenarios. +Example 2.2 (Simple read-write). Consider a very simple state, which we imagine to contain just two memory +locations (input and output), and on which we can perform the following three actions: +1. Store any value from our domain of discourse in the input location. +2. For the current value x in the input location, compute some y such that P(x, y) holds (where P is a +fixed binary predicate symbol of the logic), and store it in the output location. +3. Retrieve the computed value y from the state’s output location. +We could formalise those three actions by including two unary state predicates stored and solved, where +stored(x) denotes that x is currently stored in the input location, and solved(x) denotes that some y satisfying +P(x, y) is stores in the output location. We would then add the following axioms to ∆S, which intuitively +represent each of the above actions: +1. Γ ⊢S {α · ⊤ · stored(x)} where α ranges over all state formulas, +2. Γ ⊢S {stored(x) · ⊤ · solved(x)} +3. Γ ⊢S {solved(x) · ∃y P(x, y) · ⊤} +We can then, for example, derive the following in SL + ∆H + ∆S for ∆H = ∅, where α, β are any state +formulas: +⊢S {β · ∀x {α · ∃y P(x, y) · ⊤} · β} +An example of such a derivation is, for example: +⊢S {α · ⊤ · stored(x)} +⊢S {stored(x) · ⊤ · solved(x)} +∧SI +⊢S {α · ⊤ ∧ ⊤ · solved(x)} +∧SEL +⊢S {α · ⊤ · solved(x)} +⊢S {solved(x) · ∃y P(x, y) · ⊤} +∧SI +⊢S {α · ⊤ ∧ ∃y P(x, y) · ⊤} +∧SEL +⊢S {α · ∃y P(x, y) · ⊤} +∀SI +⊢S {β · ∀x {α · ∃y P(x, y) · ⊤} · β} +We note that while state formulas and actions are used in the proof, if we set α = β = ⊤ then the components +of the theorem itself are just formulas in ordinary first-order logic. +6 + +Example 2.3 (Fixed-length array sorting). Let us now consider our state as an array of length three, and +elements in that array as having some order structure. +We formalise this in SL by introducing 1, 2, 3 +as constants representing our three locations, along with two state predicates: a binary predicate ≤ for +comparing elements at locations l and l′, and a nullary predicate sorted that declares that the state is sorted. +These can be characterised by adding the following axiom schemes, but to ∆H rather than ∆S as they do +not represent state actions: +Γ ⊢H 1 ≤ 2 ∧ 2 ≤ 3 ⇒ sorted +Γ ⊢H l ≤ l′ ∨ l′ ≤ l +where l, l′ range over {1, 2, 3} +We then allow a single action on our array, namely the swapping of a pair of elements in the list. Suppose +that α is a state formula of the form +α :≡ l1 ≤ l′ +1 ∧ . . . ∧ ln ≤ l′ +n +(2) +where li, li range over locations {1, 2, 3}. Now for l, l′ ∈ {1, 2, 3} let α[l ↔ l′] denote α where all instances of +l and l′ are swapped, so that if e.g. α = 3 ≤ 2 ∧ 1 ≤ 2 ∧ 1 ≤ 3 then +α[2 ↔ 3] = 2 ≤ 3 ∧ 1 ≤ 3 ∧ 1 ≤ 2 +We axiomatise the swapping of the values in locations of some arbitrary pair l, l′ ∈ {1, 2, 3} by adding to ∆S +all instances of +Γ ⊢S {α · ⊤ · α[l ↔ l′]} +where α ranges over state formulas of the form (2). The statement that all arrays of length three can be +sorted is then formulated as +⊢S {⊤ · ⊤ · sorted} +Let us now give a proof of this statement in SL + ∆H + ∆S. First, let α := 1 ≤ 2 ∧ 1 ≤ 3, and define D1 as +⊢S {2 ≤ 3 ∧ α · ⊤ · 2 ≤ 3 ∧ α} +cons +⊢S {2 ≤ 3 ∧ α · ⊤ · sorted} +2↔3 +⊢S {3 ≤ 2 ∧ α · ⊤ · 2 ≤ 3 ∧ 1 ≤ 3 ∧ 1 ≤ 2} +cons +⊢S {3 ≤ 2 ∧ α · ⊤ · sorted} +cond[2≤3∨3≤2] +⊢S {α · ⊤ · sorted} +where for the left instance of cons we use 2 ≤ 3∧α ⊢H sorted, in the right that 2 ≤ 3∧1 ≤ 3∧1 ≤ 2 ⊢H sorted, +and for the final instance of cond we use ⊢H 2 ≤ 3 ∨ 3 ≤ 2. Now let D2 be defined by +1↔2 +⊢S {2 ≤ 1 ∧ 2 ≤ 3 · ⊤ · 1 ≤ 2 ∧ 1 ≤ 3} +D1 +⊢S {1 ≤ 2 ∧ 1 ≤ 3 · ⊤ · sorted} +∧SI +⊢S {2 ≤ 1 ∧ 2 ≤ 3 · ⊤ ∧ ⊤ · sorted} +∧SEL +⊢S {2 ≤ 1 ∧ 2 ≤ 3 · ⊤ · sorted} +Then we have D3: +D2 +⊢S {2 ≤ 1 ∧ 2 ≤ 3 · ⊤ · sorted} +{1 ≤ 2 ∧ 2 ≤ 3 · ⊤ · 1 ≤ 2 ∧ 2 ≤ 3} +cons +⊢S {1 ≤ 2 ∧ 2 ≤ 3 · ⊤ · sorted} +cond[2≤1∨1≤2] +⊢S {2 ≤ 3 · ⊤ · sorted} +where here cond uses ⊢H 2 ≤ 1 ∨ 1 ≤ 2, and finally +⊢S {2 ≤ 3 · ⊤ · 2 ≤ 3} +2↔3 +⊢S {3 ≤ 2 · ⊤ · 2 ≤ 3} +cond[2≤3∨3≤2] +⊢S {⊤ · ⊤ · 2 ≤ 3} +D3 +⊢S {2 ≤ 3 · ⊤ · sorted} +∧SI +⊢S {⊤ · ⊤ ∧ ⊤ · sorted} +∧SEL +⊢S {⊤ · ⊤ · sorted} +7 + +In contrast to Example 2.2 above, this is an example of a purely imperative proof that involves no proposi- +tional formulas other than ⊤. As we will see in Example 5.6, the proof corresponds to a purely imperative +program. +3 +The system ST: A simple functional/imperative term calculus +We now define our calculus ST + ΛS whose terms will represent realizers for proofs in SL + ∆H + ∆S. This +is a standard typed lambda calculus for mixed functional and imperative programs, and is defined to include +basic terms together with additional constants in some set ΛS, where the latter are intuitively there to realize +the axioms in ∆S. Semantics for the terms will be given via a monadic translation into the metalanguage +defined in the next section. Types are defined by the grammmar +X ::= D | C | X × X | X + X | X → X +while basic terms are defined as +e ::= skip | defaultX | c | f | x | p0(e) | p1(e) | e ◦ e | ι0(e) | ι1(e) | elim e e e | λx.e | e e | if α then e else e +where f ranges over all function symbols of SL, c are constants in ΛS, and α ranges over state formulas of +SL. Typing derivations of the form Γ ⊢ t : X are given below, where Γ is a set of typed variables. Note that +the types of constants c ∈ ΛS are also left unspecified. +Γ ⊢ f : Dn → D +where f has arity n +Γ ⊢ c : X +Γ ⊢ x : X +if x : X in Γ +Γ ⊢ skip : C +Γ ⊢ s : X +Γ ⊢ t : Y +Γ ⊢ s ◦ t : X × Y +Γ ⊢ t : X × Y +Γ ⊢ p0(t) : X +Γ ⊢ t : X × Y +Γ ⊢ p1(t) : Y +Γ ⊢ t : X +Γ ⊢ ι0(t) : X + Y +Γ ⊢ t : Y +Γ ⊢ ι1(t) : X + Y +Γ ⊢ r : X + Y +Γ ⊢ s : X → Z +Γ ⊢ t : Y → Z +Γ ⊢ elim r s t ⊢ Z +Γ, x : X ⊢ t : Y +Γ ⊢ λx.t : X → Y +Γ ⊢ t : X → Y +Γ ⊢ s : X +Γ ⊢ ts : Y +Γ ⊢ defaultX : X +Γ ⊢ s : X +Γ ⊢ t : X +x : D ∈ Γ for all free variables of α +Γ ⊢ if α then s else t : X +The type C should be interpreted as a type of commands that act on the state but don’t return any +values. It is be helpful to consider a derived operator for sequential composition: +Definition 3.1. If Γ ⊢ s : C and Γ ⊢ t : X then Γ ⊢ s ∗ t := p1(s ◦ t) : X. In particular, if Γ ⊢ t : C then +Γ ⊢ s ∗ t : C. +4 +A monadic embedding of SL and ST into a metatheory Sω +We now give a semantic interpretation of both state formulas of SL + ∆H + ∆S and terms in ST + ΛS into +a standard higher-order, many sorted logic Sω + ΛSω. +4.1 +The system Sω +This logic contains typed lambda terms along with equational axioms for reasoning about them, together +with the usual axioms and rules of many-sorted predicate logic. +Because most aspects of the logic are +completely standard, and in any case it is purely a verifying system, we are less detailed in specifying it. +Types are defined as follows: +X ::= D | 1 | Bool | S | X × X | X → X +8 + +where D represents objects in the domain of SL (just as in ST), Bool a type of booleans, and states are +now explicitly represented as objects of type S. Our metatheory is an equational calculus, with an equality +symbol =X for all types. Typed terms include: +• variables x, y, z, . . . for each type, where we denote state variables by π, π1, π2, . . . +• a constant f : Dn → D for each n-ary function symbol of SL, +• additional, as yet unspecified constant symbols c : X for interpreting objects in ΛS, along with axioms +that characterise them, +• a unit element () : 1 along with the axiom x = (), +• boolean constants t and f, with the axiom x =Bool t ∨ x =Bool f, +• pairing ⟨s, t⟩ and projection proj0(t), proj1(t) operators, with the usual axioms, +• terms formed by lambda abstraction and application, with the rule (λx.t)s = t[s/x], +• for each type X a case operator case (b) (s) (t) for b : Bool and s, t : X, with axioms case f x y = x and +case t x y = y. +We sometimes write xX instead of x : X, and we use abbreviations such as ⟨x, y, z⟩ for ⟨x, ⟨y, z⟩⟩. Atomic +formulas of Sω include all ordinary predicate symbols P, Q, R, . . . of SL as atomic formulas, where an n-ary +predicate P in SL takes arguments of type Dn in Sω, along with predicates p, q, r, . . . for each state predicate +symbol of SL, but now, if p is an n-ary state predicate in SL, p takes arguments of type Dn × S in Sω. +General formulas are built using the usual logical connectives, including quantifiers for all types. The axioms +and rules of Sω include the axioms of rules of predicate logic (now in all finite types), axioms for the terms, +along with the usual equality axioms (including full extensionality). Because Sω acts as a verifying theory, +we freely use strong axioms (such as extensionality), without concerning ourselves with the minimal such +system that works. +4.2 +The embedding [·] on state formulas of SL +The main purpose of our metalanguage is to allow us to reason semantically about SL and ST. To do this, +we introduce an embedding of state formulas of SL and terms of ST into Sω. We use the same notation [·] +for both, as there is no danger of ambiguity. +Definition 4.1. For each term t of SL, there is a natural interpretation of t as a term of type D in ST, namely +x �→ x : D and f(t1, . . . , tn) �→ f(t1 ◦ · · · ◦ tn) : D. Similarly, there is a natural interpretation of t into Sω, +this time with f(t1, . . . , tn) �→ f(⟨t1, . . . , tn⟩). We use the same notation for t in each of the three systems, +as there is no risk of ambiguity. +Definition 4.2. For each state formula α of SL, we define a formula [α](π) of Sω, whose free variables are +the same as those of α (but now typed with type D) with the potential addition of a single state variable π, +as follows: +• [⊤](π) := ⊤ and [⊥](π) := ⊥, +• [p(t1, . . . , tn)](π) := p(t1, . . . , tn, π), +• [α ∧ β](π) := [α](π) ∧ [β](π), and similarly for α ∨ β and α ⇒ β. +The following Lemma is easily proven using induction over propositional derivations. +Lemma 4.3. If Γ ⊢H α in SL then [α](π) is provable in Sω from the assumptions [Γ](π), where [Γ](π) := +[α1](π), . . . , [αn](π) for Γ := α1, . . . , αn. This extends to proofs in SL + ∆H provided that the embedding of +any axiom in ∆H is provable in Sω + ΛSω. +9 + +We are now in a position to make the semantic meaning of main formulas of SL precise. While, technically +speaking, this is not necessary in what follows, neither to formulate our realizability interpretation nor to +prove our soundness theorem, for the sake of completeness we give the full definition. +Definition 4.4. For each main formula A of , we define a formula [A] of Sω, whose free variables are the +same as those of A (but now typed with type D), as follows: +• [⊤] := ⊤ and [⊥] := ⊥, +• [P(t1, . . . , tn)] := P(t1, . . . , tn), +• [A ∧ B] := [A] ∧ [B], [A ∨ B] := [A] ∨ [B] and [∃x A] := ∃xD [A], +• [A ⇒ {α · B · β}] := [A] ⇒ [{α · B · β}] and [∀x {α · A · β}] := ∀xD [{α · A · β}] +where [{α · A · β}] := ∃πS [α](π) ⇒ [A] ∧ ∃π′ [β](π′). +Similarly to Lemma 4.3, we can now prove the following by induction over derivations in SL. We omit +the proof, because it is straightforward and in any case not necessary in what follows. +Proposition 4.5. If Γ ⊢S {α · A · β} in SL then [{α · A · β}] is provable in Sω from the assumptions [Γ], +where [Γ] := [A1], . . . , [An] for Γ := A1, . . . , An. This extends to proofs in SL + ∆H + ∆S provided that the +embedding of any axiom in ∆H and ∆S is provable in Sω + ΛSω. +4.3 +The embedding [·] on terms of ST +Our translation on terms is a call-by-value monadic translation using the state monad S → X × S. We first +define a translation on types of ST as follows: +• [D] := D, [C] := 1 and [X × Y ] := [X] × [Y ], +• [X + Y ] := Bool × [X] × [Y ] +• [X → Y ] := [X] → S → [Y ] × S +Lemma 4.6. For any type X of SL, the type [X] is inhabited, in the sense that we can define a canonical +closed term 0X : [X]. +Proof. Induction on types, letting 0D := c for a constant symbol which is assumed to exist in SL. The only +other nonstandard case is 0X→Y , which can be defined as λx, π . ⟨0Y , π⟩. +Finally, before introducing our translation on terms, we need to add characteristic functions to Sω +for all state formulas (analogous to the characteristic functions for quantifier-free formulas in [14]). For +any state formula α[x1, . . . , xn] of SL, where x1, . . . , xn are the free variables of α, we introduce constants +χα :Dn → S → X → X → X satisfying the axioms +[x1, . . . , xn](π) ⇒ χα ⟨x1, . . . , xn⟩ π y z = y +[¬α][x1, . . . , xn](π) ⇒ χα ⟨x1, . . . , xn⟩ π y z = z +Definition 4.7. For each term Γ ⊢ t : X of ST we define a term [Γ] ⊢ [t] : S → [X] × S of Sω as follows, +where [·] is defined on contexts as [x1 : X1, . . . , xn : Xn] := x1 : [X1], . . . , xn : [Xn]: +• [x]π := ⟨x, π⟩, +• [skip]π := ⟨(), π⟩, +• [f]π := ⟨λxDn, π . ⟨fx, π⟩, π⟩, +• [c]π is appropriately defined for each additional constant in ΛS, +10 + +• [s ◦ t]π := ⟨a, b, π2⟩ where ⟨a, π1⟩ := [s]π and ⟨b, π2⟩ := [t]π1, +• [p0t]π := ⟨a, π1⟩ and [p1t]π := ⟨b, π1⟩ where ⟨a, b, π1⟩ := [t]π, +• [ι0t]π := ⟨f, a, 0Y , π1⟩ and [ι1t] := ⟨t, 0X, b, π1⟩ for ⟨a, π1⟩ := [t]π, +• [elim r s t]π := case e (faπ2) (gbπ3) for ⟨e, a, b, π1⟩ := [r]π, ⟨f, π2⟩ := [s]π1, ⟨g, π3⟩ := [t]π1, +• [λx.t]π := ⟨λx[X].[t], π⟩, +• [ts]π := faπ2 for ⟨f, π1⟩ := [t]π and ⟨a, π2⟩ := [s]π1, +• [defaultX]π := ⟨0X, π⟩, +• [if α[x1, . . . , xn] then s else t]π := χα ⟨x1, . . . , xn⟩ π ([s]π) ([t]π) where {x1, . . . , xn} are the free variables +of α. +The following lemmas will be useful when verifying our realizability interpretation in the next section. +The first is by a simple induction on terms. +Lemma 4.8. For any term t of SL, we have [t]π = ⟨t, π⟩ (cf. Definitions 4.1 and 4.7). +Lemma 4.9 (Currying in ST). Suppose that Γ, x : X, y : Y ⊢ t : Z is a term in ST, and define Γ ⊢ λ∗v.t : +X × Y → Z by λ∗v.t := λv.(λx, y.t)(p0v)(p1v) where v is not free in t. Then for any s : X × Y we have +[(λ∗v.t)s]π = [t][a/x, b/y]π1 +where ⟨a, b, π1⟩ := [s]π. +Proof. By unwinding the definition of [·]. For any variable v : X × Y we have [p0v]π = ⟨proj0v, π⟩ and +[p1v]π = ⟨proj1v, π⟩, and we also have [λx, y . t]π = ⟨λx, π.⟨λy.[t], π⟩, π⟩. We therefore calculate +[(λx, y.t)(p0v)]π = (λx, π.⟨λy.[t], π⟩)(proj0v)π = ⟨λy.[t][proj0v/x], π⟩ +and thus +[(λx, y.t)(p0v)(p1v)]π = (λy.[t][proj0v/x])(proj1v)π = [t][proj0v/x, proj1v/y]π +Finally, we can see that if ⟨a, b, π1⟩ := [s]π then +π = (λv.[(λx, y.t)(p0v)(p1v)])(⟨a, b⟩)π1 += (λv.[t][proj0v/x, proj1v/y])(⟨a, b⟩)π1 += [t][proj0v/x, proj1v/y][⟨a, b⟩/v]π1 += [t][a/x, b/y]π1 +which completes the proof. +5 +A realizability interpretation of SL into ST +We now come to the main contribution of the paper, which is the definition of a realizability relation between +terms of ST and formulas of SL, along with a soundness theorem that shows us how to extract realizers from +proofs. Our metatheory Sω is used to define the realizability relation and prove the soundness theorem. +Definition 5.1 (Types of realizers). To each main formula A of SL we assign a type τS(A) of ST as follows: +• τS(⊤) = τS(⊥) = τS(P(t1, . . . , tn)) := C, +• τS(A ∧ B) := τS(A) × τS(B), +11 + +• τS(A ∨ B) := τS(A) + τS(B), +• τS(∃x A) := D × τS(A), +• τS(A ⇒ {α · B · β}) := τS(A) → τS(B), +• τS(∀x {α · A · β}) := D → τS(A). +Definition 5.2 (Realizability relation). For each main formula A of SL we define a formula x sr A of Sω, +whose free variables are contained in those of A (now typed with type D) together with a fresh variable +x : [τS(A)], by induction on the structure of A as follows: +• x sr Q := Q for Q = ⊤, ⊥ or P(t1, . . . , tn), +• x sr A ∧ B := (proj0x sr A) ∧ (proj1x sr B), +• x sr A ∨ B := (proj0x = f ⇒ proj0(proj1x) sr A) ∧ (proj0x = t ⇒ proj1(proj1x) sr B), +• x sr ∃y A(y) := (proj1x sr A)[proj0x/y], +• f sr (A ⇒ {α · B · β}) := ∀x[τS(A)] (x sr A ⇒ fx sr {α · B · β}), +• f sr (∀x {α(x) · A(x) · β(x)}) := ∀xD (fx sr {α(x) · A(x) · β(x)}), +where for x : S → [τS(A)] × S we define +• x sr {α · A · β} := ∀πS ([α](π) ⇒ proj0(xπ) sr A ∧ [β](proj1(xπ))). +The following substitution lemma is easily proven by induction on formulas of SL. +Lemma 5.3. For any term t of SL and s : [τS(A)] we have s sr A[t/x] = (s sr A)[t/x], where x is not free +in s and on the right hand side we implicitly mean the natural interpretation of t in Sω (cf. Definition 4.1). +Theorem 5.4 (Soundness). Suppose that +Γ := Au1 +1 , . . . , Aun +n ⊢S {α · A · β} +is provable in SL. Then we can extract from the proof a term ∆, τS(Γ) ⊢ t : τS(A) of ST, where ∆ contains +the free variables of Γ and {α · A · β} (typed with type D) and τS(Γ) := x1 : τS(A1), . . . , xn : τS(An) for fresh +variables x1, . . . , xn, such that the formula +[t] sr {α · A · β} +is provable in Sω from the assumptions (x1 sr A1)u1, . . . , (xn sr An)un for xi : [τS(Ai)]. The theorem holds +more generally for proofs in SL + ∆H + ∆S, now provably in Sω + ΛSω, if: +• for any axiom Γ ⊢H α in ∆H, the corresponding axiom [Γ](π) ⇒ [α](π) is added to ΛSω, +• for any axiom in ∆S there is a term t of ST+ΛS such that [t] realizes that axiom provably in Sω +ΛSω. +Proof. Induction on the structure of derivations in SL. +In all cases, we assume as global assumptions +(x1 sr A1)u1, . . . , (xn sr An)un, and our aim is then to produce a term t such that if [α](π) holds for some +state variable π, then a sr A and [β](π1) hold for ⟨a, π1⟩ := [t]π. +• For the axiom Γ ⊢S {α · A · α}, if Au ∈ Γ we define t := x for the corresponding variable x : τS(A). +Then [x]π := ⟨x, π⟩ for x sr A and [α](π). For Γ ⊢S {α · ⊤ · α} we define t := skip and the verification +is even simpler. +12 + +• (∧SI) Given terms s, t with [s] sr {α · A · β} and [t] sr {β · B · γ}, from [α](π) we can infer a sr A +and [β](π1) for ⟨a, π1⟩ := [s]π, and from [β](π1) it follows that b sr B and [γ](π2) for ⟨b, π2⟩ := [t]π1, +therefore we have shown that [s ◦ t] sr {α · A ∧ B · γ}. +• (∧SEi) If [t] sr {α · A ∧ B · β} then ⟨a, b⟩ sr A ∧ B and [β](π1) follow from [α](π), where ⟨a, b, π1⟩ := +[t]π. But then [p0t] sr {α · A · β} and [p1t] sr {α · B · β}. +• (∨SIi) If [t] sr {α · A · β} and [α](π) holds, then a sr A and [β](π1) for ⟨a, π1⟩ := [t]π, and therefore +(b = f ⇒ a sr A) ∧ (b = t ⇒ 0τS(B) sr B) +for b := f. Thus [ι0t] sr A ∨ B. By an entirely analogous argument we can show that [ι1t] sr A ∨ B +whenever [t] sr B. +• (∨SE) Suppose that r, s(x) and t(y) are such that [r] sr {α · A ∨ B · β}, [s](x) sr {β · C · γ} assuming +x sr A, and [t](y) sr {β · C · γ} assuming y sr B. We claim that +[elim r (λx.s) (λy.t)] sr {α · C · γ} +To prove this, first note that if [α](π), we have ⟨e, a, b⟩ sr A ∨ B and [β](π1) for ⟨e, a, b, π1⟩ := [r]π. +There are now two possibilities. If e = f then +elim r (λx.s) (λy.t)tπ = faπ2 +for ⟨f, π2⟩ := [λx.s]π1 = ⟨λx.[s](x), π1⟩ += (λx.[s](x))aπ1 += [s](a)π1 +But since [β](π1) holds and e = f also implies that a sr A, we have c sr C and [γ](π2) for ⟨c, π2⟩ := +[s](a)π1, which proves the main claim in the case e = f. An analogous argument works for the case +e = t. +• (⇒S I) If t(x) is such that [t](x) sr {α · B · β} whenever x sr A, then by definition we have +λx.[t] sr A ⇒ {α · B · β} +and therefore [λx.t] sr {γ · A ⇒ {α · B · β} · γ} for any γ. +• (⇒S E) Assume that [s] sr {β · A · γ} and [t] sr {α · A ⇒ {γ · B · δ} · β}. If [α](π) holds then defining +⟨f, π1⟩ := [t]π we have [β]π1 and +x sr A ⇒ fx sr {γ · B · δ} +Similarly, defining ⟨a, π2⟩ := [s]π1, it follows that [γ](π2) and a sr A. Finally, setting ⟨b, π3⟩ := faπ2 it +follows that b sr B and [δ](π3), and we have therefore proven that [ts] sr {α · B · δ}. +• (⊥SE) If [t] sr {α · ⊥ · β} then from [α](π) we can infer a sr ⊥ and [β](π2) for ⟨a, π1⟩ := [t]π. But +a sr ⊥ = ⊥, and from ⊥ we can deduce anything, and in particular 0τS(A) sr A and [γ](π), from which +it follows that [defaultτS(A)] sr {α · A · γ}. +• (∀SI) Suppose that t(x) is such that [t](y) sr {α[y/x] · A[y/x] · β[y/x]}, where y ≡ x or y is not free in +{α · A · β}, and y is not free in Γ. Then since y is not free in any of the assumptions xi sr Ai, we can +deduce in Sω that +∀xD [t](x) sr {α · A · β} +and therefore λx.[t] sr ∀x {α · A · β}, and thus (just as for ⇒S I) we have +[λx.t] sr {γ · ∀x {α · A · β} · γ} +for any γ. +13 + +• (∀SE) Suppose that [s] sr {α · ∀x {β · A · γ} · β[t/x]} and that [α](π) holds. Then f sr ∀x {β · A · γ} +and [β][t/x](π1) for ⟨f, π⟩ := [s]π. +Now, using Lemma 4.8 we have [st]π = ftπ1 for the natural +interpretation of t in Sω, since we can prove in Sω that +ft sr {β[t/x] · A[t/x] · γ[t/x]} +it follows that a sr A[t/x] and [γ][t/x](π2) for ⟨a, π2⟩ := ftπ1, and therefore we have shown that +[st] sr {α · A[t/x] · γ[t/x]}. +• (∃SI) If [s] sr {α · A[t/x] · β} and [α](π) then a sr A[t/x] and [β](π1) for ⟨a, π1⟩ := [s]π. By Lemma +5.3 we therefore have (a sr A)[t/x], and therefore ⟨t, a⟩ sr ∃x A. Observing (using Lemma 4.8) that +[t ◦ s]π = ⟨t, a, π1⟩, we have shown that [t ◦ s] sr {α · ∃x A · β}. +• (∃SE) Suppose that s and t(x, z) are such that [s] sr {α · ∃x A · β} and +z sr A[y/x] ⇒ [t](y, z) sr {β · C · γ} +where y ≡ x or y is not free in A, and y is also not free in C, α, β, γ or Γ. By Lemma 5.3 that +z sr A[y/x] = (z sr A)[y/x] = ⟨y, z⟩ sr ∃x A we therefore have +⟨y, z⟩ sr ∃x A ⇒ [t](y, z) sr {β · C · γ} +Now, applying Lemma 4.9 to ∆, Γ, y : D, z : τS(A) ⊢ t : τS(C), we have +[(λ∗v.t)s]π = [t](e, a)π1 +for ⟨e, a, π1⟩ := [s]π. Now, if [α](π) holds, then we have ⟨e, a⟩ sr ∃x A and [β](π1), and therefore since +[t](e, a) sr {β · C · γ}, we have c sr C and [γ](π2) for ⟨c, π2⟩ = [t](e, a)π1 = [(λ∗v.t)s]π, and thus we +have shown that [(λ∗v.t)s] sr {α · C · γ}. +• (cons) If α ⊢H β and γ ⊢H δ then by Lemma 4.3 both [α](π) ⇒ [β](π) and [γ](π) ⇒ [δ](π) are provable +in Sω (respectively Sω + ΛSω for the general version of the theorem) for any π : S. It is then easy to +show that if [t] sr {β · A · γ} then we also have [t] sr {α · A · δ}. +• (cond) Suppose that [s] sr {α ∧ γ · A · δ} and [t] sr {β ∧ γ · A · δ}. We claim that +[if α then s else t] sr {γ · A · δ} +To prove this, suppose that [γ](π) holds. Since ⊢H α ∨ β then [α](π) ∨ [β](π) is provable in Sω, and so +we consider two cases. Let {x1, . . . , xn} be the free variables of α. If [α](π) holds, then +[if α then s else t]π = χα ⟨x1, . . . , xn⟩ π ([s]π) ([t]π) = [s]π +and since then [α](π) ∧ [γ](π) we have a sr A and [δ](π1) for ⟨a, π1⟩ := [s]π. On the other hand, if +[β](π) holds, then by an analogous argument we can show that a sr A and [δ](π1) for ⟨a, π1⟩ := [t]π = +[if α then s else t]π, and we are done. +The extension of the soundness theorem to SL + ∆H + ∆S is straightforward, as the soundness proof is +modular and so any axioms along with their realizers can be added. The first condition is needed so that +Lemma 4.3 (needed for the cons rule) continues to apply. +For the free variable condition that the free variables of t are contained in those of Γ, {α · A · β} and +τS(Γ), if this were not the case, we could simply ground those variables with a canonical constant c : D and +we would still have ˜t sr {α · A · β} for the resulting term ˜t. +14 + +Corollary 5.5 (Program extraction). Suppose that the sentence +⊢S {α · ∀x {β · ∃y P(x, y) · γ(x)} · β} +is provable in SL + ∆S. Then we can extract a closed realizing term t : D → D × C in ST + ΛS such that +defining g : D → S → D × S by gxπ := ⟨a, π2⟩ for ⟨f, π1⟩ := [t]π and ⟨a, (), π2⟩ := fxπ1, we have +∀πS([α](π) ⇒ ∀xD (P(x, proj0(gxπ)) ∧ [γ](x)(proj1(gxπ)))) +provably in Sω + ΛSω. +5.1 +Simplification and removal of unit types +In presentations of modified realizability that use product types instead of type sequences, it is common to +introduce the notion of a Harrop formula (a formula that does not contain disjunction or existential quantifi- +cation in a positive position) and define realizability in a way that all Harrop formulas have unit realizability +type, so that e.g. τS(∀x (P ∧ Q)) = 1 for atomic predicates P and Q, rather than τS(∀x (P ∧ Q)) = D → 1×1 +as for us. We have avoided this simplification earlier on, as it would have added additional cases and bureau- +cracy to our soundness theorem. However, we can compensate retroactively for this choice by introducing +equivalences on types that eliminate unit types, namely the closure under contexts of +1 × X ≃ 1 ≃ X × 1 +(1 → X) ≃ X +(X → 1) ≃ 1 +along with corresponding equivalences on terms, also closed under contexts: +t1×X ≃ proj1(t)X +tX×1 ≃ proj0(t)X +t1→X ≃ t() +tX ≃ λx1.t +tX→1 ≃ () +For example, in Corollary 5.5 we would then have +[t]π : (D → S → D × 1 × S) × S ≃ (D → S → D × S) × S +and +gxπ ≃ fxπ1. +For us, the equivalence relation ≃ will not play a formal role in the paper, but will be used to provide +simplified descriptions of extracted programs. +5.2 +Examples of program extraction +We now continue the short illustrative examples we outlined in Section 2.3. +Example 5.6 (Simple read-write). In Example 2.2 we considered a state where three actions were possible +(writing to the state, performing a calculation, and reading the output from the state). We can formalise these +three actions semantically in the metatheory Sω by including three constants in ΛSω, namely c1 : D → S → S, +c2 : S → S and c3 : S → D, along with the characterising axioms: +1. stored(x, c1xπ), +2. stored(x, π) ⇒ solved(x, c2π), +3. solved(x, π) ⇒ P(x, c3π). +While we are able to use these constants to form terms in Sω such as λπ, π1, x . ⟨c1xπ, c2π1⟩, which could be +viewed as non-sequential in the sense that we take two input states as arguments, we can force them to be +applied in a sequential, call-by-value manner by adding three corresponding constants to our term calculus +ST, namely including write : D → C, calc : C and read : D × C in ΛS, along with the embedding rules +• [write]π := ⟨λx, π′ . ⟨(), c1xπ′⟩, π⟩ ≃ ⟨c1, π⟩, +• [calc]π := ⟨(), c2π⟩ so that [calc] ≃ c2, +15 + +• [read]π := ⟨c3π, (), π⟩ ≃ ⟨c3π, π⟩. +and then restricting out attention to terms of the form [t] for t ∈ ST + {write, calc, read}. We can then prove +the following in Sω i.e. that all axioms in ∆S can be realised: +• [write(x)] sr {α · ⊤ · stored(x)}, +• [calc] sr {stored(x) · ⊤ · solved(x)}, +• [read] sr {solved(x) · ∃y P(x, y) · ⊤}. +and thus Theorem 5.4 applies to SL + ∆H + ∆S for ∆H = ∅. In particular, we have +[t] sr {β · ∀x {α · ∃y P(x, y) · ⊤} · β} +for t := λx . ((write(x) ∗ calc) ∗ read) where ∗ is sequential composition operator from Definition 3.1. A +formal derivation of this term from the corresponding proof given in Example 2.2 is as follows: +x : D ⊢ write(x) : C +x : D ⊢ calc : C +∧SI +x : D ⊢ write(x) ◦ calc : C × C +∧SEL +x : D ⊢ write(x) ∗ calc : C +x : D ⊢ read : D × C +∧SI +x : D ⊢ (write(x) ∗ calc) ◦ read : C × D × C +∧SEL +x : D ⊢ (write(x) ∗ calc) ∗ read : D × C +∀SI +⊢ λx . ((write(x) ∗ calc) ∗ read) : D → D × C +Example 5.7 (Fixed-length array sorting). In Example 2.3 we considered a situation where we are allowed a +single action on our state, namely to swap elements. Analogously to the previous example, we can formalise +this in our semantic environment Sω by adding to ΛSω constants cl,l′ : S → S for each pair l, l′ ∈ {1, 2, 3} +along with the axiom +[α](π) ⇒ [α[l ↔ l′]](cl,l′π) +ranging over state formulas α of the form (2) and locations l, l′ ∈ {1, 2, 3} of SL, together with axioms +corresponding to those of ∆H i.e. +(π) ⇒ sorted(π) +and +[l ≤ l′ ∨ l′ ≤ l](π) +Similarly, for each l, l′ ∈ {1, 2, 3} we add a term swapl,l′ : C to ΛS and define [swapl,l′]π := ⟨(), cl,l′π⟩ so that +swapl,l′ sr {α · ⊤ · α[l ↔ l′]} +A derivation of a closed term t : C of ST + {swapl,l′} such that [t] sr {⊤ · ⊤ · sorted} is given below. In +particular, we can prove in Sω that ∀πS sorted(proj1([t]π)), and so the term λπ . proj1([t]π) : S → S acts as +a sorting program for arrays of length three. For an extracted term t corresponding to the proof given in +Example 2.3, first we interpret D1 as +⊢ skip : C +cons +⊢ skip : C +2↔3 +⊢ swap2,3 : C +cons +⊢ swap2,3 : C +cond[2≤3∨3≤2] +⊢ t1 := if (2 ≤ 3) then (skip) else (swap2,3) : C +and define t1 := if (2 ≤ 3) then (skip) else (swap2,3). Now D2 is interpreted as +1↔2 +⊢ swap1,2 : C +D1 +⊢ t1 : C +∧SI +⊢ swap1,2 ◦ t1 : C × C +∧SEL +⊢ t2 := swap1,2 ∗ t1 : C +16 + +where we define t2 := swap1,2 ∗ t1 : C. Continuing, D3 is interpreted as: +D2 +⊢ t2 : C +⊢ skip : C +cons +⊢ skip : C +cond[2≤1∨1≤2] +t3 := if (2 ≤ 1) then t2 else (skip) : C +where t3 := if (2 ≤ 1) then t2 else (skip), and finally +⊢ skip : C +2↔3 +⊢ swap2,3 : C +cond[2≤3∨3≤2] +⊢ if (2 ≤ 3) then (skip) else (swap2,3) : C +D3 +⊢ t3 : C +∧SI +⊢ (if (2 ≤ 3) then (skip) else (swap2,3)) ◦ t3 : C × C +∧SEL +⊢ t := (if (2 ≤ 3) then (skip) else (swap2,3)) ∗ t3 : C +6 +An extension to arithmetic +We now present an extension of our framework to a stateful version of first-order intuitionistic arithmetic. +On the logic side, we will add not only a stateful induction rule, but also a Hoare-style while rule for iteration +over the natural numbers. On the computational side, these will be interpreted by stateful recursion in all +finite types, along with a controlled while loop. The addition of these constants will allow us to extract +programs that are more interesting than those obtainable from proofs in pure predicate logic, and which +can be clearly compared to well-known stateful algorithms. To exemplify this, we will present a formally +synthesised version of insertion sort, and we stress that by further extending our framework with additional +rules and terms, we would be able to extract an even richer variety of combined functional/stateful programs. +6.1 +The system SA: First-order arithmetic with state +Our system of stateful intuitionistic arithmetic SA builds on SL just as ordinary first-order Heyting arithmetic +builds on first-order predicate logic. In both cases, we introduce a constant 0, a unary successor symbol succ, +symbols for all primitive recursive functions, and our predicate symbols now include an equality relation =. +In what follows we write x + 1 instead of succ(x). The axioms and rules of SA are, in turn, analogous to +the additional axioms and rules we would require in ordinary first-order arithmetic: They include all axioms +and rules of SL (based now on the language of SA), along with a collection of additional axioms and rules. +These comprise not only basic axioms and rules for equality and the successor, and an induction rule (all now +adapted to incorporate the state), but also a new while rule for stateful iteration, which now exploits our +state and, as we will see, allows us to extract programs that contain while loops. These additional axioms +and rules are outlined in Figure 6.1. +Our formulation of stateful arithmetic follows the same basic idea as the construction of stateful predicate +logic, incorporating standard rules but keeping track of an ambient state in a call-by-value manner, and +adding new rules that explicitly correspond to stateful constructions. In particular, Proposition 2.1 clearly +extends to SA, as the usual axioms and rules of arithmetic can be embedded into those of SA: +Proposition 6.1. For any formula A of HA and state formula α, define the main formula Aα of SA as in +Proposition 2.1. Then whenever Γ ⊢I A is provable in HA, we have that Γα, ∆ ⊢S {α · Aα · α} is provable in +SA, where ∆ is arbitrary and Γα := (A1)u1 +α , . . . , (An)un +α +for Γ := Au1 +1 , . . . , Aun +n . +We can also derive a natural extensionality rule from our stateful equality rules, which assures us that +whenever s = t in ordinary Heyting arithmetic, then we can replace s by t for stateful formulas: +Proposition 6.2. Suppose that ⊢I s = t is provable in HA. Then from Γ ⊢S {α(s) · A(s) · β(s)} we can +derive Γ ⊢S {α(t) · A(t) · β(t)} in SA. +17 + +Figure 3: Additional axioms and rules of SA +Axioms and rules for equality +Γ ⊢S {α · t = t · α} +Γ ⊢S {α · s = t · β} +Γ ⊢S {α · t = s · β} +Γ ⊢S {α · r = s · β} +Γ ⊢S {β · s = t · γ} +Γ ⊢S {α · r = t · γ} +Γ ⊢S {α · s = t · β} +Γ ⊢S {β · A(s) · γ(s)} +ext +Γ ⊢S {α · A(t) · γ(t)} +Axioms and rules for arithmetical function symbols +Γ ⊢S {α · succ(t) ̸= 0 · α} +{α · succ(s) = succ(t) · β} +{α · s = t · β} +Γ ⊢S {α · l = r · α} +where l = r ranges across defining equations for prim. rec. functions +Induction rule +Γ ⊢S {α · A(0) · β(0)} +Γ, A(x) ⊢S {β(x) · A(x + 1) · β(x + 1)} +ind +Γ ⊢S {γ · ∀x {α · A(x) · β(x)} · γ} +While rule (over natural numbers) +A1 +A2 +A3 +while +Γ, A(x) ⊢S {α(x) · B · β} +A1 := Γ, A(x + 1) ⊢S {γ(x + 1) ∧ α(x + 1) · A(x) · α(x)} +A2 := Γ, A(x + 1) ⊢S {¬γ(x + 1) ∧ α(x + 1) · B · β} +A3 := Γ, A(0) ⊢S {α(0) · B · β} +for ind and while, x is not free in Γ, and for while it is not free in B or β +Proof. By Proposition 6.1 for α := α(s) we have Γ ⊢S {α(s) · s = t · α(s)} and thus using the extensionality +rule in SA we can derive +Γ ⊢S {α(s) · s = t · α(s)} +Γ ⊢S {α(s) · A(s) · β(s)} +ext +Γ ⊢S {α(s) · A(t) · β(t)} +Since ⊢I t = s must also be provable in HA, another instance of Proposition 6.1 for α := α(t) along with the +true axiom in SA gives us +Γ ⊢S {α(t) · t = s · α(t)} +Γ ⊢S {α(t) · ⊤ · α(t)} +ext +Γ ⊢S {α(t) · ⊤ · α(s)} +Putting these together we obtain +Γ ⊢S {α(t) · ⊤ · α(s)} +Γ ⊢S {α(s) · A(t) · β(t)} +∧SI +Γ ⊢S {α(t) · ⊤ ∧ A(t) · β(t)} +∧SEL +Γ ⊢S {α(t) · A(t) · β(t)} +which completes the derivation. +18 + +6.2 +An extended term calculus STN +In order to give derivations in SA a computation interpretation, we need to extend our term calculus ST +to include a recursor (for induction) and a controlled while loop (for the while rule). The remaining new +axioms and rules of SA are dealt with in a straightforward manner. +To be precise: the theory STN is defined to be the instance of ST for the case of arithmetic, with function +symbols for zero, successor and all primitive recursive functions. Accordingly, we rename the base type D +to Nat. In addition to the terms of ST, we add terms rec e e and whilee γ[z]e e e to our grammar, where γ[z] +ranges over state formulas of SL with a specified free variable z. The typing rules for these new terms are +Γ ⊢ s : X +Γ ⊢ t : Nat → X → X +Γ ⊢ rec s t : Nat → X +for the recursor, while for the while loop we have +Γ ⊢ r : Nat → X → X +Γ ⊢ s : Nat → X → Y +Γ ⊢ t : X → Y +Γ ⊢ u : Nat +Γ ⊢ whileu γ[z] r s t : X → Y +under the additional variable condition that z /∈ Γ, but x : Nat ∈ Γ for all free variables of γ[z] outside of +z. Note that we do not consider z a free variable of whilea γ[z] r s t, but rather a placeholder for the loop +condition. In order to give the appropriate semantics to our terms, we must add to our metatheory Sω +axioms and rules for arithmetic in all finite types, including the ability to define functions of arbitrary type +via recursion over the natural numbers, along the lines of E-HAω [30] (though as before the precise details +are not important). We then define: +• [rec s t]π := ⟨Rf, π1⟩ for ⟨f, π1⟩ := [t]π, where +Rf0π := [s]π +Rf(n + 1)π := gaπ′ +2 for ⟨a, π′ +1⟩ := Rfnπ′ and ⟨g, π′ +2⟩ := fnπ′ +1 +(3) +• [whileu γ[z] r s t]π := ⟨Lf,g,hm, π4⟩ where ⟨f, π1⟩ := [r]π, ⟨g, π2⟩ := [s]π1, ⟨h, π3⟩ := [t]π2 and ⟨m, π4⟩ := +[u]π3, where +Lf,g,h0yπ′ := hyπ′ +Lf,g,h(n + 1)yπ′ := +� +Lf,g,hny′π2 for ⟨a, π′ +1⟩ := fnπ′ and ⟨y′, π′ +2⟩ := ayπ′ +1 +if [γ][n + 1](π′) +byπ′ +1 for ⟨b, π′ +1⟩ := gnπ′ +if ¬[γ][n + 1](π′) +(4) +where in the case distinctions, we would technically speaking need to use the characteristic function +χγ⟨x1, . . . , n, . . . , xk⟩ for γ, with n substituted for the special free variable z. +6.3 +The soundness theorem for arithmetic +We now need to show that the soundness proof for stateful predicate logic also holds in the extension to +arithmetic. +Theorem 6.3. The statement of Theorem 5.4 remains valid if we replace SL by SA and ST by STN. +Proof. We need to extend the proof of Theorem 6.3 to show that the additional axioms and rules as in Figure +6.1 can be realized by a term of the form [t] for t in STN. +• For the non-extensionality equality and arithmetic axioms this is straightforward due to the fact that +these are also true in Sω: For instance, given a realizer [s] sr {α · u = v · β} and [t] sr {β · v = w · γ}, +we have that [s ◦ t] sr {α · u = v ∧ v = w · γ}, and since from u = v ∧ v = w we can infer u = w in Sω, +it follows that [p1(s ◦ t)] sr {α · u = w · γ}. The other axioms and rules are even simpler. +19 + +• (ext) Extensionality is similarly simple: If [s] sr {α · u = v · β} and [t] sr {β · A(u) · γ(u)}, then [α](π) +implies that u = v and [β](π1) for ⟨. . . , π1⟩ := [s]π, and therefore a sr A(u) and [γ](u)(π2) for +⟨a, π2⟩ := [t]π1. Now applying extensionality in Sω to the formula T (x) := a sr A(x) ∧ [γ](x)(π2), +from u = v we have a sr A(v) and [γ](v)(π2), and thus [s ◦ t] sr {α · u = v ∧ A(v) · γ(v)} and therefore +[p2(s ◦ t)] sr {α · A(v) · γ(v)}. +• (rec) Suppose that s and t(x, y) are such that [s] sr {α · A(0) · β(0)} and +[t](x, y) sr {β(x) · A(x + 1) · β(x + 1)} +assuming y sr A(x). We show that [rec s λx, y.t(x, y)] sr {γ · ∀x {α · A(x) · β(x)} · γ} for any γ. Since +[rec s λx, y.t(x, y)]π = ⟨Rf, π⟩ for f := λx.[λy.t(x, y)] and Rf as in (3), it suffices to show that for any +n : Nat we have +Rfn sr {α · A(n) · β(n)} +We prove this by induction: For the base case, we have Rf0 = [s] and the claim holds by assumption. +For the induction step, let us assume that [α](π′) holds, and so by the induction hypothesis we have +a sr A(n) and [β(n)](π′ +1) for ⟨a, π′ +1⟩ := Rfn. Since fnπ′ +1 = ⟨g, π′ +1⟩ for g := λy.[t](n, y), we have that +Rf(n+1)π′ = [t](n, a)π′ +1, and since by the property of [t] we then have b sr A(n + 1) and [β(n+1)](π′ +2) +for ⟨b, π′ +2⟩ := [t](n, a)π′ +1, we have shown that Rf(n + 1) sr {α · A(n + 1) · β(n + 1)}, which completes +the induction. +• (while) We suppose that +1. [r](x, y) sr {γ(x + 1) ∧ α(x + 1) · A(x) · α(x)} assuming that y sr A(x + 1), +2. [s](x, y) sr {¬γ(x + 1) ∧ α(x + 1) · B · β} assuming that y sr A(x + 1), +3. [t](y) sr {α(0) · B · β} assuming that y sr A(0). +Our aim is to show that +[(whilex γ (λx′, y′.r) (λx′, y′.s) (λy′.t))y] sr {α(x) · B · β} +for any x, y ∈ Nat with y sr A(x). We observe, unwinding the definition, that +[(whilex γ (λx′, y′.r) (λx′, y′.s) (λy′.t))y]π = Lf,g,hxyπ +for f := λx′.[λy′.r(x′, y′)], g := λx′.[λy′.s(x′, y′)], h := λy′.[t](y′) and Lf,g,h as defined in (4). We now +show by induction on n that if y sr A(n) then +Lf,g,hny sr {α(n) · B · β} +and then the result follows by setting n := x. The base case is straightforward since +Lf,g,y0y = [t](y) +and the claim follows by definition of [t]. For the induction step, suppose that y sr A(n + 1) and +[α(n + 1)](π). There are two cases. If ¬[γ](n + 1)(π) we have +Lf,g,h(n + 1)yπ = [s](n, y)π +and the result holds by the property of [s]. On the other hand, if [γ](n + 1)(π) then +Lf,g,h(n + 1)yπ = Lf,g,hny′π′ +for ⟨y′, π′⟩ := [r](n, y)π. But by the property of [r] we have y′ sr A(n) and [α(n)](π′), and therefore by +the induction hypothesis we have b sr B and [β](π′′) for ⟨b, π′′⟩ := Lf,g,hny′π′ = Lf,g,h(n + 1)yπ, and +so the result is proven for n + 1. +This covers all the additional axioms and rules of SA. +20 + +6.4 +Worked example: Insertion sort +We now illustrate our extended system by synthesising a list sorting program that, intuitively, forms an +implementation of the insertion sort algorithm. Here our state will represent the structure that is to be +sorted, and continuing the spirit of generality that we have adhered to throughout, we characterise this +structure through a number of abstract axioms. Instantiating the state as, say, an array of natural numbers, +would provide a model for our theory, but our sorting algorithm can be extracted on the more abstract level. +Crucially, the proof involves both loop iteration and induction, and the corresponding program combines an +imperative while loop with a functional recursor. +We begin by axiomatising our state, just as in previous examples. An intuition here is that states represent +an infinite array of elements a0, a1, . . . possessing some total order structure ≤, and we seek to extract a +program that, for any input n, sorts the first n elements. We use this informal semantics throughout to +indicate the intended meaning of our axioms, but stress that none of this plays a formal role in the proof or +resulting computational interpretation. +We introduce three state predicates to SA, with the intuition indicated in each case: +• sort(N) +Sorted: The first N + 1 elements of the array i.e. [a0, . . . , aN] are sorted +• psort(n, N) +Partially sorted with respect to an: if n < N then the list [a0, . . . , an−1, an+1, . . . , aN] is sorted and +an ≤ an+1. For the base cases, if n = N then the list [a0, . . . , aN−1] is sorted, and if n > N then the +list [a0, . . . , aN] is sorted. +• comm(n) +Comparison: true if an ≤ an−1, and always true if n = 0 +We formalise this intuition by adding the following state independent axioms to ∆H: +1. Γ, sort(N) ⊢H psort(N + 1, N + 1) +If the first N +1 elements are sorted, then they are also partially sorted with respect to the next element +aN+1. +2. Γ, ¬comm(n), psort(n, N) ⊢H sort(N) +If [a0, . . . , an−1, an+1, . . . , aN] is sorted, an ≤ an+1, but also an−1 ≤ an, then the entire segment +[a0, . . . , aN] must be sorted. +3. Γ, psort(0, N) ⊢H sort(N) +If [a1, . . . , aN] is sorted and a0 ≤ a1, then [a0, . . . , aN] is sorted. +4. Γ ⊢H sort(0) +The singleton array [a0] is defined to be sorted. +We complete the axiomatisation by adding a single state-sensitive axiom to ∆S: +5 Γ ⊢S {comm(n + 1) ∧ psort(n + 1, N) · ⊤ · psort(n, N)} +If [a0, . . . , an, an+2, . . . , aN] is sorted and an+1 ≤ an+2, but an+1 ≤ an, then we can modify the state +(i.e. swapping an and an+1 by setting ˜an := an+1 and ˜an+1 := an) so that [a0, . . . , an−1, ˜an+1, . . . , aN] +is sorted and ˜an ≤ ˜an+1. The edge cases for n ≥ N are interpreted in a more straightforward way. +In order to give a realizing term to this axiom, we representing element swapping semantically by adding a +constant c : Nat → S → S to our metatheory Sω, which satisfies +comm(n + 1, π) ∧ psort(n + 1, N, π) ⇒ psort(n, N, cnπ) +21 + +and a corresponding term swap : Nat → C to our term calculus, along with the embedding +[swap]π := ⟨λn, π.⟨(), cnπ⟩, π⟩ ≃ ⟨c, π⟩ +so that we can prove +[swap n] sr {comm(n + 1) ∧ psort(n + 1, N) · ⊤ · psort(n, N)} +With this in place, we can now prove in SA that the first N elements of the state can be sorted, and extract +a corresponding realizing term in STN. +6.4.1 +Proof of ⊢S {γ · ∀N {α · ⊤ · sort(N)} · γ} in SA +The core of our proof begins with an instance of the while rule parametrised by N, with Γ := ∅, A(n) := ⊤, +α(n) := psort(n, N + 1), β := sort(N + 1) and γ(n) := comm(n): +D1 +D2 +D3 +while +⊤ ⊢S {psort(n, N + 1) · ⊤ · sort(N + 1)} +∀SI +⊤ ⊢S {psort(N + 1, N + 1) · ∀n {psort(n, N + 1) · ⊤ · sort(N + 1)} · psort(N + 1, N + 1)} +∀SE +⊤ ⊢S {psort(N + 1, N + 1) · ⊤ · sort(N + 1)} +cons +⊤ ⊢S {sort(N) · ⊤ · sort(N + 1)} +where the final composition inference makes use of the first state independent axiom. Here D1 represents an +instance of the state sensitive axiom +⊤ ⊢S {comm(n + 1) ∧ psort(n + 1, N + 1) · ⊤ · psort(n, N + 1)} +and D2 represents the derivation +⊤ ⊢S {sort(N + 1) · ⊤ · sort(N + 1)} +cons +⊤ ⊢S {¬comm(n + 1) ∧ psort(n + 1, N + 1) · ⊤ · sort(N + 1)} +where composition makes use of the second state independent axiom. Finally D3 is +⊤ ⊢S {sort(N + 1) · ⊤ · sort(N + 1)} +cons +⊤ ⊢S {psort(0, N + 1) · ⊤ · sort(N + 1)} +this time making use of the third state independent axiom. Finally we can prove that all lists can be sorted +with an outer induction as follows: +⊢S {α · ⊤ · α} +cons +⊢S {α · ⊤ · sort(0)} +D +⊤ ⊢S {sort(N) · ⊤ · sort(N + 1)} +ind +⊢S {γ · ∀N {α · ⊤ · sort(N)} · γ} +where α is an arbitrary state predicate, the instance of cons uses the fourth state independent axiom, and +D represents the derivation above. +6.4.2 +Program extraction +We now extract a program that corresponds to the above proof. First of all, we note that the three premises +of our while rule are realised by swap n, skip and skip respectively, and so our derivation D corresponds to +the following program: +y : C ⊢ swap n : C +y : C ⊢ skip : C +y : C ⊢ skip : C +while +y : C ⊢ t(n)y : C +∀SI +y : C ⊢ λn.t(n)y : Nat → C +∀SE +y : C ⊢ (λn.t(n)y)(N + 1) : C +cons +y : C ⊢ (λn.t(n)y)(N + 1) : C +22 + +where +t(n) := whilen comm[z] (λx, y.(swap x)) (λx, y.skip) (λy.skip) +≃ whilen comm[z] (λx.(swap x)) (skip) (skip) +Then our final induction generates the following program: +⊢ skip : C +y : C ⊢ (λn.t(n)y)(N + 1) : C +ind +⊢ rec (skip) (λx, y.((λn.t(n)y)(x + 1))) : Nat → C +Thus our list sorting program is +rec (skip) (λx, y.((λn.t(n)y)(x + 1))) +≃ rec (skip) (λx.((λn.(whilen comm[z] (λx.(swap x)) (skip) (skip)()))(x + 1))) +which is essentially an implementation of the insertion sort algorithm, with an outer recursion that sorts +initial segments of the list in turn, and an inner loop that inserts new elements into the appropriate place in +the current sorted list. +7 +Directions for future work +In this paper we have presented the central ideas behind a new method for extracting stateful programs +from proofs, which include an extension of ordinary first-order logic with Hoare triples, a corresponding +realizability interpretation, and a soundness theorem. We emphasise once again that our intention has been +to offer an alternative approach to connecting proofs with stateful programs, one that seeks to complement +rather than improve existing work by embracing simplicity and abstraction, and which might be well suited +to a range of applications in proof theory or computability theory. In this spirit, we conclude with a very +informal outline of a series interesting directions in which we anticipate that our framework could be applied. +7.1 +Further extensions and program synthesis +While our main results have been presented in the neutral setting of first-order predicate logic, it would be +straightforward to extend SL to richer logics with more complex data structures and a imperative commands. +Already, the addition of recursion and loops over natural numbers in Section 6 has allowed us to synthesise +a standard in-place sorting algorithm using our abstract axiomatisation of an ordered state, in a similar +spirit to [6]. However, further extensions are naturally possible, including the addition of general fixpoint +operators and non-controlled while loops, which would then require a Sω to be replaced by a domain theoretic +semantics that allows for partiality. +Looking a step further ahead, by implementing all of this in a proof assistant, we would have at our +disposal a new technique for synthesising correct-by-construction imperative programs. While we do not +suggest that this pipeline would directly compete with existing techniques for verifying imperative programs, +it could be well suited to synthesising and reasoning about programs in very specific domains, where we are +interested in algorithms for which interactions with the state have a restricted form that could be suitably +axiomatised within our logic. For example, a more detailed axiomatisation our state as an ordered array along +the lines of Section 6.4, with a “swap” operation and a few other ways of interacting with the state, might +give rise to an interesting theory of in-place sort algorithms. Stateful algorithms on other data structures, +such as graphs, could presumably also be formalised within our framework. +7.2 +Bar recursion and the semantics of extracted programs +Two of the main starting points for this paper, the monadic realizability of Birolo [7] and the author’s own +Dialectica interpretation with state [24], address the broader problem of trying to understand the operational +semantics of programs extracted from proofs as stateful procedures (the origins and development of this +general idea, from Hilbert’s epsilon calculus onwards, is brilliantly elucidated in Chapter 1 of Aschieri’s +23 + +thesis [2], who then sets out his own realizability interpretation based on learning). A number of case studies +by the author and others [20, 21, 25, 26] have demonstrated that while terms extracted from nontrivial proofs +can be extremely complex, they are often much easier to understand if one focuses on the way they interact +with the mathematical environment. For example, in understanding a program extracted from a proof using +Ramsey’s theorem for pairs [20], it could be illuminating to study the trace of the program as it queries a +colouring at particular pairs, as this can lead to a simpler characterisation of the algorithm ultimately being +implemented by the term. +While the aforementioned analysis of programs has always been done in an informal way, our stateful +realizability interpretation would in theory allow us to extract programs which store this trace formally in +the state, where our abstract characterisation of state would allow us to implement it in whichever way is +helpful in a given setting. For example, in the case of the Bolzano-Weierstrass theorem [21], our state might +record information of the form xn ∈ I, collecting information about the location of sequence elements. For +applications in algebra [26], one might instead store information about a particular maximal ideal. +The aforementioned theorems are typically proven using some form of choice or comprehension, and that +in itself leads to the interesting prospect of introducing both stateful recursors and while-loops that are +computationally equivalent to variants of bar recursion [29]. In [23], several bar recursive programs that +arise from giving a computational interpretation to arithmetical comprehension principles are formulated as +simple while loops, and these could in principle be incorporated into our system with new controlled Hoare +rules in the style of update recursion [5], that replace the conditions n < N and n ≥ N in the Ai above with +e.g. n ∈ dom(f) and n /∈ dom(f), where f is some partial approximation to a comprehension function. An +exploration of such while-loops from the perspective of higher-order computability theory might well be of +interest in its own right. +7.3 +A logic for probabilistic lambda calculi +Probabilistic functional languages are a major topic of research at present. While work in this direction dates +back to the late 1970s [16, 28] where it typically had a semantic flavour, a more recent theme [9, 11, 12] has +been to study simple extensions of the lambda calculus with nondeterministic choice operators ⊕, where s⊕t +evaluates nondeterministically (or probabilistically) to either s or t. While such calculi have been extensively +studied, corresponding logics that map under some proof interpretation to probabilistic programs are far +more rare (although there is some recent work in this direction e.g. [1]). +We conjecture that our framework offers a bridge between logic and probabilistic computation through +incorporating probabilistic disjunctions into our logic SL and taking states to be streams of outcomes of +probabilistic events together with a current ‘counter’ that increases each time an event occurs. In a simple +setting where only two outcomes are possible with equal probability, we can axiomatise this within SL by +adding zero and successor functions (allowing us to create numerals n), along with a unary state predicate +count(n). We can then model probabilistic events by adding the appropriate axioms to ∆S. Suppose, for +example, we add two predicate constants H(x) and T (x) (for heads and tails), along with constants c1, c2, . . . +representing coins. Then flipping a coin would be represented by the axiom schema +Γ ⊢S {count(n) · H(ci) ∨ T (ci) · count(n + 1)} +where n ranges over numerals and ci over coin constants, the counter indicating that a probabilistic event has +occurred. The act of reading a probability from the state could be interpreted semantically by introducing +a constant ω : S → Bool × S to Sω, with the axiom +count(n, π) ⇒ (e = f ⇒ H(ci)) ∧ (e = t ⇒ T (ci)) ∧ count(n + 1, π1) for ⟨e, π1⟩ := ωπ +(alternatively, we could simply define S := Nat × (Nat → Bool) for a type of Nat natural numbers, and +define ω⟨n, a⟩ := ⟨a(n), ⟨n + 1, a⟩⟩ and count(n, ⟨m, a⟩) := m =Nat n). +A probabilistic choice operator ⊕ can then be added to the language of ST, along with the typing rule +Γ ⊢ s ⊕ t : X + Y for Γ ⊢ s : X and Γ ⊢ t : Y , and the interpretation +[s ⊕ t]π := case e ([ι0s]π1) ([ι1t]π1) where ⟨e, π1⟩ := ωπ +24 + +In particular, defining flip := skip ⊕ skip : C + C we would have +[flip] sr {count(n) · H(ci) ∨ T (ci) · count(n + 1)} +although we stress that the operator ⊕ and would allow for much more complex probabilistic disjunctions, +potentially involving additional computational content. +Our soundness theorem, extended to these new probabilistic axioms and terms, would then facilitate the +extraction of probabilistic programs from proofs. For instance, including a winner predicate W(x), two player +constant symbols p1, p2, and adding axioms H(c1), H(c2) ⊢S {α · W(p1) · α}; T (c1), T (c2) ⊢S {α · W(p1) · α}; +H(c1), T (c2) ⊢S {α · W(p2) · α} and T (c1), H(c2) ⊢S {α · W(p2) · α} for any α, we could prove +⊢S {count(n) · ∃x W(x) · count(n + 2)} +expressing the fact that a winner can be determined after two flips. We can then extract a corresponding +probabilistic term for realizing this statement, which would be isomorphic to the expected program that +queries the state twice in order to determine the outcome of those flips, and returns either p1 or p2 as a +realizer for ∃x W(x) depending on the content of the state. +Of course, the details here need to be worked through carefully in order to properly substantiate the +claim that our framework could be used to extract probabilistic programs in a natural and meaningful way. +At the very least, it is likely that further additions to SL along with a more intricate state would be needed +to incorporate more interesting probabilistic events, such as annotated disjunctions along the lines of [31]. +We leave such matters to future work. +References +[1] Antonelli, M., Dal Lago, U., and Pistone, P. Curry and Howard meet Borel. Preprint, 2022. +[2] Aschieri, F. Learning, Realizability and Games in Classical Arithmetic. 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In Proceedings of Foundations of Software Science and Computation Structures (FOSSACS’07) +(2007), vol. 44213 of LNCS, pp. 361–377. +27 + diff --git a/29AzT4oBgHgl3EQfuf0F/content/tmp_files/load_file.txt b/29AzT4oBgHgl3EQfuf0F/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa4eb19ae49520ae92501404ed98a7d4d2b35a83 --- /dev/null +++ b/29AzT4oBgHgl3EQfuf0F/content/tmp_files/load_file.txt @@ -0,0 +1,1022 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf,len=1021 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='01690v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='LO] 4 Jan 2023 Proofs as stateful programs: A first-order logic with abstract Hoare triples, and an interpretation into an imperative language Thomas Powell January 5, 2023 Abstract We introduce an extension of first-order logic that comes equipped with additional predicates for reasoning about an abstract state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Sequents in the logic comprise a main formula together with pre- and postconditions in the style of Hoare logic, and the axioms and rules of the logic ensure that the assertions about the state compose in the correct way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The main result of the paper is a realizability interpretation of our logic that extracts programs into a mixed functional/imperative language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' All programs expressible in this language act on the state in a sequential manner, and we make this intuition precise by interpreting them in a semantic metatheory using the state monad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Our basic framework is very general, and our intention is that it can be instantiated and extended in a variety of different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We outline in detail one such extension: A monadic version of Heyting arithmetic with a wellfounded while rule, and conclude by outlining several other directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 1 Introduction The Curry-Howard correspondence lies at the heart of theoretical computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Over the years, a multitude of different techniques for extracting programs from proofs have been developed, the majority of which translate formal proof systems into lambda calculi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' As such, programs extracted from proofs are typically conceived as pure functional programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Everyday programmers, on the other hand, often think and write in an imperative paradigm, in terms of instructions that change some underlying global state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' This is reinforced by the fact that many of the most popular programming languages, including C and Python, lean towards this style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Imperative programs are nevertheless highly complex from a mathematical perspective, and while systems such as Hoare logic [15] or separation logic [27] have been designed to reason about them, the formal extraction of imperative programs from proofs has received comparatively little attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' In this paper, we propose a new idea in this direction, developing a formal system SL that enriches ordinary first-order logic with Hoare triples for reasoning about an abstract global state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Sequents will have the form Γ ⊢ {α · A · β}, where A is a formula and α, β assertions about the state, and proofs in the logic will include both ordinary introduction and elimination rules for predicate logic, together with special rules for reasoning about the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We then construct a stateful realizability interpretation (based on Kreisel’s modified realizability [17]) that relates formulas in SL to terms in a mixed functional/imperative language ST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Our main result is a soundness theorem, which confirms that whenever a formula is provable in SL, we can extract a corresponding stateful realizing term in ST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' While our initial soundness theorem focuses on pure predicate logic, we subsequently show that it can be extended to arithmetic, where in particular we are then able to extract programs that contain both recursion and controlled while loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We are not the first to adapt traditional methods to extract imperative programs: A major achievement in this direction, for example, is the monograph [22], which sets up a variant of intuitionistic Hoare logic alongside a realizability translation into a standard imperative language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Other relevant examples include [3, 8, 10, 13, 19, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' However, these and almost all other prior work in this direction tend to focus on formal verification, with an eye towards using proof interpretations as a method for the synthesis of correct- by-construction software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' In concrete terms, this means that the formal systems tend to be quite detailed 1 and oriented towards program analysis, while the starting point is typically a program for which we want to construct a verification proof, rather than a proof from which we hope to extract a potentially unfamiliar program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Our approach, on the other hand, is much more abstract, with an emphasis on potential applications in logic and proof theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Our basic system SL makes almost no assumptions about the structure of the state and what we are allowed to do with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Rather, we focus on producing a general framework for reasoning about ‘stateful formulas’, which can then be instantiated with additional axioms to model concrete scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The simplicity and generality of our framework is its most important feature, and we consider this work to be a first step towards a number of potentially interesting applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For this reason, we include not only an extension of our system to a monadic theory of arithmetic, but conclude by sketching out some additional ways in which we conjecture that our logic and interpretation could be used and expanded, including the computational semantics of proofs and probabilistic logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We take ideas from three main sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The first is a case study of Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [6], in which a realizability interpretation is used to extract a version of in-place quicksort, and where the imperative nature of the extracted program is presented in a semantic way using the state monad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' While their program behaves imperatively “by-chance”, terms extracted from our logic are forced to be imperative, and thus our framework offers one potential solution to their open problem of designing a proof calculus which only yields imperative programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Indeed, an implementation of the insert sort algorithm is formally extracted in Section 6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Our second source of inspiration is the thesis of Birolo [7], where a general monadic realizability interpretation is defined and then used to give an alternative, semantic presentation of learning- based interactive realizability [2, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' However, our work goes beyond this in that it also involves a monadic extension of the target logic, whereas Birolo’s applies to standard first-order logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Finally, a number of ideas are taken from the author’s previous work [24] on extracting stateful programs using the Dialectica interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' While there the state is used in a very specific and restricted way, unlike our more general presentation here, we use an analogous call-by-value monadic translation on terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' It is important to stress that we do not claim that our work represents an optimal or complete method for extracting imperative programs from proofs, nor do we claim that it is superior to alternative methods, including the aforementioned works in the direction of verification, or, for instance, techniques based on Krivine’s classical realizability [18], which could be viewed as imperative in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We simply offer what we consider to be a new and interesting perspective that emphasises abstraction and simplicity, and propose that our framework could prove valuable in a number of different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Overview of the paper The main technical work that follows involves the design of three different systems, a realizability interpre- tation that connects them, and an instantiation of this framework in the setting of first-order arithmetic, namely: A novel extension SL of predicate logic with abstract Hoare triples, which can be extended with additional axioms for characterising the state (Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' A standard calculus ST for lambda terms with imperative commands, which can again be extended with additional constants for interacting with the state (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' A metalanguage Sω into which both SL and ST can be embedded (Section 4), which is used to formulate the realizability relation and prove its soundness (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' An instantiation of SL as a theory of arithmetic, with programs extracted into an extension of ST with recursion and while loops (Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Concrete examples are given, and potential applications surveyed in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 2 2 The system SL: First-order logic with state We begin by introducing our target theory SL from which stateful programs will be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' This is an extension of ordinary first-order logic in the sense that the latter can always be embedded into SL (we will make this precise in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Ultimately, we are interested not so much in SL on its own, but in theories of the form SL + ∆H + ∆S, where ∆H and ∆S are collections of (respectively non-computational and computational) axioms that together characterise the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Several concrete examples will be given to illustrate this, and in Section 6 we present a variant of SL that represents a theory of first-order arithmetic with state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1 Intuitionistic first-order logic Before defining SL, we give a standard presentation of first-order intuitionistic predicate logic PL, which serves as an opportunity to fix our basic style of formal reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The language of PL consists of the logical constants ∧, ∨, ⇒, ∀, ∃, ⊤, ⊥, variables x, y, z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=', along with function symbols f, g, h, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' and predicate symbols P, Q, R, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=', each with a fixed arity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We assume the existence of at least one constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Terms are built from variables and function symbols as usual, and formulas are built from prime formulas P(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , tn), ⊤ and ⊥ using the logical constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We use the usual abbreviation ¬A :≡ A ⇒ ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We work in a sequent style natural deduction calculus, where sequents have the form Γ ⊢I A for some context Γ and formula A, and a context is a set of labelled assumptions of the form Au1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , Aun n for pairwise distinct labels ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The axioms and rules of PL are as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Figure 1: Axioms and rules of PL Propositional logic Γ ⊢I A if Au ∈ Γ for some u Γ ⊢I ⊤ Γ ⊢I A Γ ⊢I B ∧I Γ ⊢I A ∧ B Γ ⊢I A ∧ B ∧EL Γ ⊢I A Γ ⊢I A ∧ B ∧ER Γ ⊢I B Γ ⊢I A ∨IL Γ ⊢I A ∨ B Γ ⊢I B ∨IR Γ ⊢I A ∨ B Γ ⊢I A ∨ B Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Au ⊢I C Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Bv ⊢I C ∨E Γ ⊢I C Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Au ⊢I B ⇒I Γ ⊢I A ⇒ B Γ ⊢I A ⇒ B Γ ⊢I A ⇒E Γ ⊢I B Γ ⊢I ⊥ ⊥E Γ ⊢I A Quantifier rules Γ ⊢I A[y/x] ∀I Γ ⊢I ∀xA Γ ⊢I ∀xA ∀E Γ ⊢I A[t/x] Γ ⊢I A[t/x] ∃I Γ ⊢I ∃xA Γ ⊢I ∃xA Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' A[y/x]u ⊢I C ∃E Γ ⊢I C for ∀I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' y ≡ x or y not free in A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' and y not free in Γ for ∃E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' y ≡ x or y not free in A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' and y not free in C or Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='2 Stateful first-order logic We now define our new logical system SL, which is an extension of ordinary first-order logic with new state propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' To be more precise, we extend the language of PL with a ternary operation {−·−·−}, together with special state predicate symbols p, q, r, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=', which also have a fixed arity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Terms of SL are the same as those of PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' On the other hand, there are two kinds of formulas in SL: state formulas and main formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' A state formula is defined using state predicate symbols and propositional connectives as follows: ⊤ and ⊥ are state formulas, if p a state predicate symbol of arity n and t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , tn are terms, then p(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , tn) is a state formula, if α, β are state formulas, so are α ∧ β, α ∨ β, α ⇒ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' A main formula (or just formula) of SL is now defined as: ⊤ and ⊥ are formulas, if P is an ordinary predicate symbol of arity n and t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , tn are terms, then P(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , tn) is a formula, if A, B are formulas, so are A ∧ B, A ∨ B and ∃xA, if A, B are formulas and α, β state formulas, then A ⇒ {α · B · β} and ∀x {α · A · β} are formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The notions of free and bound variables, along with substitution α[t/x] and A[t/x] can be easily defined for both state and main formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Analogous to the construction of formulas, our basic proof system uses the auxiliary notion of a state proof in order to define a main proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' A state sequent has the form Γ ⊢H α where α is a state formula and Γ a set of labelled state formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' A proof of Γ ⊢H α in SL is built from the axioms and rules of classical propositional logic i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' the propositional axioms and rules as set out in Figure 1 plus the law of excluded middle Γ ⊢H α ∨ ¬α, together with a set ∆H of as yet unspecified state axioms of the form Γ ⊢H α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' A main sequent of SL has the form Γ ⊢S {α · A · β}, where A is a formula and α, β state formulas, and Γ is a set of labelled main formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' A proof of Γ ⊢S {α · A · β} in SL uses the axioms and rules given in Figure 2, together with a set ∆S of additional axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We now make precise what we mean when we characterise SL as an extension of standard first-order logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The following is provable with an easy induction over derivations in PL: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For any formula A of PL and state formula α, define the main formula Aα of SL by Qα := Q for Q atomic, (A ∧ B)α := Aα ∧ Bα, (A ∨ B)α := Aα ∨ Bα and (∃x A)α := ∃x Aα, (A ⇒ B)α := Aα ⇒ {α · Bα · α} and (∀x A)α := ∀x {α · Aα · α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Then whenever Γ ⊢I A is provable in PL, we have that Γα, ∆ ⊢S {α · Aα · α} is provable in SL, where ∆ is arbitrary and Γα := (A1)u1 α , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , (An)un α for Γ := Au1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , Aun n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3 The intuition behind SL The intended semantic meaning of Γ ⊢H α is that α can be inferred from the assumptions Γ for any fixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' More specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' if we imagine a semantic variant [α](π) of each state formula where now the dependency on an underlying state π is made explicit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' the semantics of Γ ⊢H α is just [Γ](π) ⇒ [α](π) 4 Figure 2: Axioms and rules of SL Propositional axioms and rules Γ ⊢S {α · A · α} if Au ∈ Γ for some u Γ ⊢S {α · ⊤ · α} Γ ⊢S {α · A · β} Γ ⊢S {β · B · γ} ∧SI Γ ⊢S {α · A ∧ B · γ} Γ ⊢S {α · A ∧ B · β} ∧SEL Γ ⊢S {α · A · β} Γ ⊢S {α · A ∧ B · β} ∧SER Γ ⊢S {α · B · β} Γ ⊢S {α · A · β} ∨SIL Γ ⊢S {α · A ∨ B · β} Γ ⊢S {α · B · β} ∨SIR Γ ⊢S {α · A ∨ B · β} Γ ⊢S {α · A ∨ B · β} Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Au ⊢S {β · C · γ} Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Bv ⊢S {β · C · γ} ∨SE Γ ⊢S {α · C · γ} Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Au ⊢S {α · B · β} ⇒SI Γ ⊢S {γ · A ⇒ {α · B · β} · γ} Γ ⊢S {α · A ⇒ {γ · B · δ} · β} Γ ⊢S {β · A · γ} ⇒SE Γ ⊢S {α · B · δ} Γ ⊢S {α · ⊥ · β} ⊥SE Γ ⊢S {α · A · γ} Quantifier rules Γ ⊢S {α[y/x] · A[y/x] · β[y/x]} ∀SI Γ ⊢S {γ · ∀x {α · A · β} · γ} Γ ⊢S {α · ∀x {β · A · γ} · β[t/x]} ∀SE Γ ⊢S {α · A[t/x] · γ[t/x]} Γ ⊢S {α · A[t/x] · β} ∃SI Γ ⊢S {α · ∃xA · β} Γ ⊢S {α · ∃xA · β} Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' A[y/x]u ⊢S {β · C · γ} ∃SE Γ ⊢S {α · C · γ} for ∀SI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' y ≡ x or y not free in A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' and y not free in Γ for ∃SE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' y ≡ x or y not free in A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' and y not free in C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' γ or Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Basic Hoare rules α ⊢H β Γ ⊢S {β · A · γ} γ ⊢H δ cons Γ ⊢S {α · A · δ} ⊢H α ∨ β Γ ⊢S {α ∧ γ · A · δ} Γ ⊢S {β ∧ γ · A · δ} cond Γ ⊢S {γ · A · δ} Additional axioms state axioms ∆H of the form Γ ⊢H α main axioms ∆S of the form Γ ⊢S {α · A · β} 5 On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' the intended meaning of Γ ⊢S {α · A · β} is that from assumptions Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' if α holds with respect to some initial state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' then we can infer that A is true and β holds with respect to some modified state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' or more precisely: [Γ] ⇒ (∃π [α](π) ⇒ ([A] ∧ ∃π′ [β](π′))) (1) In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' the computational interpretation of (1) above will be a program that takes some input state π satisfying [α](π) and returns a realizer-state pair ⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' π′⟩ such that x realizes A and [β](π′) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Our semantic interpretation [·] will be properly defined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Crucially, in SL the state is implicit, and so there are no variables or terms of state type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The state will rather be made explicit in our metatheory Sω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The main axioms and rules of SL simply describe how this semantic interpretation propagates in a call-by-value manner through the usual axioms and rules of first-order logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The state itself is brought into play through the Hoare rules along with the additional axioms ∆H and ∆S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The two Hoare rules of SL correspond to the consequence and conditional rules of traditional Hoare logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The usual conditional rule falls out as a special case of ours since we assume Γ ⊢H α ∨ ¬α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Some of the other traditional Hoare rules are derivable: The empty statement axiom corresponds to our own axiom Γ ⊢S {α · ⊤ · α} while composition can be viewed as the special case of ∧SI for A = B = ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' In Section 6 we add a controlled while loop to our logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' But for now, we illustrate our logic with some very straightforward scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='2 (Simple read-write).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Consider a very simple state, which we imagine to contain just two memory locations (input and output), and on which we can perform the following three actions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Store any value from our domain of discourse in the input location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For the current value x in the input location, compute some y such that P(x, y) holds (where P is a fixed binary predicate symbol of the logic), and store it in the output location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Retrieve the computed value y from the state’s output location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We could formalise those three actions by including two unary state predicates stored and solved, where stored(x) denotes that x is currently stored in the input location, and solved(x) denotes that some y satisfying P(x, y) is stores in the output location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We would then add the following axioms to ∆S, which intuitively represent each of the above actions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Γ ⊢S {α · ⊤ · stored(x)} where α ranges over all state formulas, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Γ ⊢S {stored(x) · ⊤ · solved(x)} 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Γ ⊢S {solved(x) · ∃y P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' y) · ⊤} We can then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' derive the following in SL + ∆H + ∆S for ∆H = ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' where α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' β are any state formulas: ⊢S {β · ∀x {α · ∃y P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' y) · ⊤} · β} An example of such a derivation is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' for example: ⊢S {α · ⊤ · stored(x)} ⊢S {stored(x) · ⊤ · solved(x)} ∧SI ⊢S {α · ⊤ ∧ ⊤ · solved(x)} ∧SEL ⊢S {α · ⊤ · solved(x)} ⊢S {solved(x) · ∃y P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' y) · ⊤} ∧SI ⊢S {α · ⊤ ∧ ∃y P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' y) · ⊤} ∧SEL ⊢S {α · ∃y P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' y) · ⊤} ∀SI ⊢S {β · ∀x {α · ∃y P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' y) · ⊤} · β} We note that while state formulas and actions are used in the proof,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' if we set α = β = ⊤ then the components of the theorem itself are just formulas in ordinary first-order logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 6 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3 (Fixed-length array sorting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Let us now consider our state as an array of length three, and elements in that array as having some order structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We formalise this in SL by introducing 1, 2, 3 as constants representing our three locations, along with two state predicates: a binary predicate ≤ for comparing elements at locations l and l′, and a nullary predicate sorted that declares that the state is sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' These can be characterised by adding the following axiom schemes, but to ∆H rather than ∆S as they do not represent state actions: Γ ⊢H 1 ≤ 2 ∧ 2 ≤ 3 ⇒ sorted Γ ⊢H l ≤ l′ ∨ l′ ≤ l where l, l′ range over {1, 2, 3} We then allow a single action on our array, namely the swapping of a pair of elements in the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Suppose that α is a state formula of the form α :≡ l1 ≤ l′ 1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' ∧ ln ≤ l′ n (2) where li, li range over locations {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Now for l, l′ ∈ {1, 2, 3} let α[l ↔ l′] denote α where all instances of l and l′ are swapped, so that if e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' α = 3 ≤ 2 ∧ 1 ≤ 2 ∧ 1 ≤ 3 then α[2 ↔ 3] = 2 ≤ 3 ∧ 1 ≤ 3 ∧ 1 ≤ 2 We axiomatise the swapping of the values in locations of some arbitrary pair l, l′ ∈ {1, 2, 3} by adding to ∆S all instances of Γ ⊢S {α · ⊤ · α[l ↔ l′]} where α ranges over state formulas of the form (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The statement that all arrays of length three can be sorted is then formulated as ⊢S {⊤ · ⊤ · sorted} Let us now give a proof of this statement in SL + ∆H + ∆S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' First, let α := 1 ≤ 2 ∧ 1 ≤ 3, and define D1 as ⊢S {2 ≤ 3 ∧ α · ⊤ · 2 ≤ 3 ∧ α} cons ⊢S {2 ≤ 3 ∧ α · ⊤ · sorted} 2↔3 ⊢S {3 ≤ 2 ∧ α · ⊤ · 2 ≤ 3 ∧ 1 ≤ 3 ∧ 1 ≤ 2} cons ⊢S {3 ≤ 2 ∧ α · ⊤ · sorted} cond[2≤3∨3≤2] ⊢S {α · ⊤ · sorted} where for the left instance of cons we use 2 ≤ 3∧α ⊢H sorted, in the right that 2 ≤ 3∧1 ≤ 3∧1 ≤ 2 ⊢H sorted, and for the final instance of cond we use ⊢H 2 ≤ 3 ∨ 3 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Now let D2 be defined by 1↔2 ⊢S {2 ≤ 1 ∧ 2 ≤ 3 · ⊤ · 1 ≤ 2 ∧ 1 ≤ 3} D1 ⊢S {1 ≤ 2 ∧ 1 ≤ 3 · ⊤ · sorted} ∧SI ⊢S {2 ≤ 1 ∧ 2 ≤ 3 · ⊤ ∧ ⊤ · sorted} ∧SEL ⊢S {2 ≤ 1 ∧ 2 ≤ 3 · ⊤ · sorted} Then we have D3: D2 ⊢S {2 ≤ 1 ∧ 2 ≤ 3 · ⊤ · sorted} {1 ≤ 2 ∧ 2 ≤ 3 · ⊤ · 1 ≤ 2 ∧ 2 ≤ 3} cons ⊢S {1 ≤ 2 ∧ 2 ≤ 3 · ⊤ · sorted} cond[2≤1∨1≤2] ⊢S {2 ≤ 3 · ⊤ · sorted} where here cond uses ⊢H 2 ≤ 1 ∨ 1 ≤ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' and finally ⊢S {2 ≤ 3 · ⊤ · 2 ≤ 3} 2↔3 ⊢S {3 ≤ 2 · ⊤ · 2 ≤ 3} cond[2≤3∨3≤2] ⊢S {⊤ · ⊤ · 2 ≤ 3} D3 ⊢S {2 ≤ 3 · ⊤ · sorted} ∧SI ⊢S {⊤ · ⊤ ∧ ⊤ · sorted} ∧SEL ⊢S {⊤ · ⊤ · sorted} 7 In contrast to Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='2 above, this is an example of a purely imperative proof that involves no proposi- tional formulas other than ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' As we will see in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='6, the proof corresponds to a purely imperative program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 3 The system ST: A simple functional/imperative term calculus We now define our calculus ST + ΛS whose terms will represent realizers for proofs in SL + ∆H + ∆S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' This is a standard typed lambda calculus for mixed functional and imperative programs, and is defined to include basic terms together with additional constants in some set ΛS, where the latter are intuitively there to realize the axioms in ∆S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Semantics for the terms will be given via a monadic translation into the metalanguage defined in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Types are defined by the grammmar X ::= D | C | X × X | X + X | X → X while basic terms are defined as e ::= skip | defaultX | c | f | x | p0(e) | p1(e) | e ◦ e | ι0(e) | ι1(e) | elim e e e | λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='e | e e | if α then e else e where f ranges over all function symbols of SL, c are constants in ΛS, and α ranges over state formulas of SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Typing derivations of the form Γ ⊢ t : X are given below, where Γ is a set of typed variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Note that the types of constants c ∈ ΛS are also left unspecified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Γ ⊢ f : Dn → D where f has arity n Γ ⊢ c : X Γ ⊢ x : X if x : X in Γ Γ ⊢ skip : C Γ ⊢ s : X Γ ⊢ t : Y Γ ⊢ s ◦ t : X × Y Γ ⊢ t : X × Y Γ ⊢ p0(t) : X Γ ⊢ t : X × Y Γ ⊢ p1(t) : Y Γ ⊢ t : X Γ ⊢ ι0(t) : X + Y Γ ⊢ t : Y Γ ⊢ ι1(t) : X + Y Γ ⊢ r : X + Y Γ ⊢ s : X → Z Γ ⊢ t : Y → Z Γ ⊢ elim r s t ⊢ Z Γ, x : X ⊢ t : Y Γ ⊢ λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t : X → Y Γ ⊢ t : X → Y Γ ⊢ s : X Γ ⊢ ts : Y Γ ⊢ defaultX : X Γ ⊢ s : X Γ ⊢ t : X x : D ∈ Γ for all free variables of α Γ ⊢ if α then s else t : X The type C should be interpreted as a type of commands that act on the state but don’t return any values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' It is be helpful to consider a derived operator for sequential composition: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' If Γ ⊢ s : C and Γ ⊢ t : X then Γ ⊢ s ∗ t := p1(s ◦ t) : X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' In particular, if Γ ⊢ t : C then Γ ⊢ s ∗ t : C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 4 A monadic embedding of SL and ST into a metatheory Sω We now give a semantic interpretation of both state formulas of SL + ∆H + ∆S and terms in ST + ΛS into a standard higher-order, many sorted logic Sω + ΛSω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1 The system Sω This logic contains typed lambda terms along with equational axioms for reasoning about them, together with the usual axioms and rules of many-sorted predicate logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Because most aspects of the logic are completely standard, and in any case it is purely a verifying system, we are less detailed in specifying it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Types are defined as follows: X ::= D | 1 | Bool | S | X × X | X → X 8 where D represents objects in the domain of SL (just as in ST), Bool a type of booleans, and states are now explicitly represented as objects of type S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Our metatheory is an equational calculus, with an equality symbol =X for all types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Typed terms include: variables x, y, z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' for each type, where we denote state variables by π, π1, π2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' a constant f : Dn → D for each n-ary function symbol of SL, additional, as yet unspecified constant symbols c : X for interpreting objects in ΛS, along with axioms that characterise them, a unit element () : 1 along with the axiom x = (), boolean constants t and f, with the axiom x =Bool t ∨ x =Bool f, pairing ⟨s, t⟩ and projection proj0(t), proj1(t) operators, with the usual axioms, terms formed by lambda abstraction and application, with the rule (λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t)s = t[s/x], for each type X a case operator case (b) (s) (t) for b : Bool and s, t : X, with axioms case f x y = x and case t x y = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We sometimes write xX instead of x : X, and we use abbreviations such as ⟨x, y, z⟩ for ⟨x, ⟨y, z⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Atomic formulas of Sω include all ordinary predicate symbols P, Q, R, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' of SL as atomic formulas, where an n-ary predicate P in SL takes arguments of type Dn in Sω, along with predicates p, q, r, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' for each state predicate symbol of SL, but now, if p is an n-ary state predicate in SL, p takes arguments of type Dn × S in Sω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' General formulas are built using the usual logical connectives, including quantifiers for all types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The axioms and rules of Sω include the axioms of rules of predicate logic (now in all finite types), axioms for the terms, along with the usual equality axioms (including full extensionality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Because Sω acts as a verifying theory, we freely use strong axioms (such as extensionality), without concerning ourselves with the minimal such system that works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='2 The embedding [·] on state formulas of SL The main purpose of our metalanguage is to allow us to reason semantically about SL and ST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' To do this, we introduce an embedding of state formulas of SL and terms of ST into Sω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We use the same notation [·] for both, as there is no danger of ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For each term t of SL, there is a natural interpretation of t as a term of type D in ST, namely x �→ x : D and f(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , tn) �→ f(t1 ◦ · · · ◦ tn) : D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Similarly, there is a natural interpretation of t into Sω, this time with f(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , tn) �→ f(⟨t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , tn⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We use the same notation for t in each of the three systems, as there is no risk of ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For each state formula α of SL, we define a formula [α](π) of Sω, whose free variables are the same as those of α (but now typed with type D) with the potential addition of a single state variable π, as follows: [⊤](π) := ⊤ and [⊥](π) := ⊥, [p(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , tn)](π) := p(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , tn, π), [α ∧ β](π) := [α](π) ∧ [β](π), and similarly for α ∨ β and α ⇒ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The following Lemma is easily proven using induction over propositional derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' If Γ ⊢H α in SL then [α](π) is provable in Sω from the assumptions [Γ](π), where [Γ](π) := [α1](π), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , [αn](π) for Γ := α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' This extends to proofs in SL + ∆H provided that the embedding of any axiom in ∆H is provable in Sω + ΛSω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 9 We are now in a position to make the semantic meaning of main formulas of SL precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' While, technically speaking, this is not necessary in what follows, neither to formulate our realizability interpretation nor to prove our soundness theorem, for the sake of completeness we give the full definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For each main formula A of , we define a formula [A] of Sω, whose free variables are the same as those of A (but now typed with type D), as follows: [⊤] := ⊤ and [⊥] := ⊥, [P(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , tn)] := P(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , tn), [A ∧ B] := [A] ∧ [B], [A ∨ B] := [A] ∨ [B] and [∃x A] := ∃xD [A], [A ⇒ {α · B · β}] := [A] ⇒ [{α · B · β}] and [∀x {α · A · β}] := ∀xD [{α · A · β}] where [{α · A · β}] := ∃πS [α](π) ⇒ [A] ∧ ∃π′ [β](π′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Similarly to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3, we can now prove the following by induction over derivations in SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We omit the proof, because it is straightforward and in any case not necessary in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' If Γ ⊢S {α · A · β} in SL then [{α · A · β}] is provable in Sω from the assumptions [Γ], where [Γ] := [A1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , [An] for Γ := A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' This extends to proofs in SL + ∆H + ∆S provided that the embedding of any axiom in ∆H and ∆S is provable in Sω + ΛSω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3 The embedding [·] on terms of ST Our translation on terms is a call-by-value monadic translation using the state monad S → X × S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We first define a translation on types of ST as follows: [D] := D, [C] := 1 and [X × Y ] := [X] × [Y ], [X + Y ] := Bool × [X] × [Y ] [X → Y ] := [X] → S → [Y ] × S Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For any type X of SL, the type [X] is inhabited, in the sense that we can define a canonical closed term 0X : [X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Induction on types, letting 0D := c for a constant symbol which is assumed to exist in SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The only other nonstandard case is 0X→Y , which can be defined as λx, π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' ⟨0Y , π⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Finally, before introducing our translation on terms, we need to add characteristic functions to Sω for all state formulas (analogous to the characteristic functions for quantifier-free formulas in [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For any state formula α[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , xn] of SL, where x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , xn are the free variables of α, we introduce constants χα :Dn → S → X → X → X satisfying the axioms [x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , xn](π) ⇒ χα ⟨x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , xn⟩ π y z = y [¬α][x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , xn](π) ⇒ χα ⟨x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , xn⟩ π y z = z Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For each term Γ ⊢ t : X of ST we define a term [Γ] ⊢ [t] : S → [X] × S of Sω as follows, where [·] is defined on contexts as [x1 : X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , xn : Xn] := x1 : [X1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , xn : [Xn]: [x]π := ⟨x, π⟩, [skip]π := ⟨(), π⟩, [f]π := ⟨λxDn, π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' ⟨fx, π⟩, π⟩, [c]π is appropriately defined for each additional constant in ΛS, 10 [s ◦ t]π := ⟨a, b, π2⟩ where ⟨a, π1⟩ := [s]π and ⟨b, π2⟩ := [t]π1, [p0t]π := ⟨a, π1⟩ and [p1t]π := ⟨b, π1⟩ where ⟨a, b, π1⟩ := [t]π, [ι0t]π := ⟨f, a, 0Y , π1⟩ and [ι1t] := ⟨t, 0X, b, π1⟩ for ⟨a, π1⟩ := [t]π, [elim r s t]π := case e (faπ2) (gbπ3) for ⟨e, a, b, π1⟩ := [r]π, ⟨f, π2⟩ := [s]π1, ⟨g, π3⟩ := [t]π1, [λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t]π := ⟨λx[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [t], π⟩, [ts]π := faπ2 for ⟨f, π1⟩ := [t]π and ⟨a, π2⟩ := [s]π1, [defaultX]π := ⟨0X, π⟩, [if α[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , xn] then s else t]π := χα ⟨x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , xn⟩ π ([s]π) ([t]π) where {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , xn} are the free variables of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The following lemmas will be useful when verifying our realizability interpretation in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The first is by a simple induction on terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For any term t of SL, we have [t]π = ⟨t, π⟩ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Definitions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='9 (Currying in ST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Suppose that Γ, x : X, y : Y ⊢ t : Z is a term in ST, and define Γ ⊢ λ∗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t : X × Y → Z by λ∗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t := λv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (λx, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t)(p0v)(p1v) where v is not free in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Then for any s : X × Y we have [(λ∗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t)s]π = [t][a/x, b/y]π1 where ⟨a, b, π1⟩ := [s]π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' By unwinding the definition of [·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For any variable v : X × Y we have [p0v]π = ⟨proj0v, π⟩ and [p1v]π = ⟨proj1v, π⟩, and we also have [λx, y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' t]π = ⟨λx, π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='⟨λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [t], π⟩, π⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We therefore calculate [(λx, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t)(p0v)]π = (λx, π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='⟨λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [t], π⟩)(proj0v)π = ⟨λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [t][proj0v/x], π⟩ and thus [(λx, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t)(p0v)(p1v)]π = (λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [t][proj0v/x])(proj1v)π = [t][proj0v/x, proj1v/y]π Finally, we can see that if ⟨a, b, π1⟩ := [s]π then π = (λv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [(λx, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t)(p0v)(p1v)])(⟨a, b⟩)π1 = (λv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [t][proj0v/x, proj1v/y])(⟨a, b⟩)π1 = [t][proj0v/x, proj1v/y][⟨a, b⟩/v]π1 = [t][a/x, b/y]π1 which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 5 A realizability interpretation of SL into ST We now come to the main contribution of the paper, which is the definition of a realizability relation between terms of ST and formulas of SL, along with a soundness theorem that shows us how to extract realizers from proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Our metatheory Sω is used to define the realizability relation and prove the soundness theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1 (Types of realizers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' To each main formula A of SL we assign a type τS(A) of ST as follows: τS(⊤) = τS(⊥) = τS(P(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , tn)) := C, τS(A ∧ B) := τS(A) × τS(B), 11 τS(A ∨ B) := τS(A) + τS(B), τS(∃x A) := D × τS(A), τS(A ⇒ {α · B · β}) := τS(A) → τS(B), τS(∀x {α · A · β}) := D → τS(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='2 (Realizability relation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For each main formula A of SL we define a formula x sr A of Sω, whose free variables are contained in those of A (now typed with type D) together with a fresh variable x : [τS(A)], by induction on the structure of A as follows: x sr Q := Q for Q = ⊤, ⊥ or P(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , tn), x sr A ∧ B := (proj0x sr A) ∧ (proj1x sr B), x sr A ∨ B := (proj0x = f ⇒ proj0(proj1x) sr A) ∧ (proj0x = t ⇒ proj1(proj1x) sr B), x sr ∃y A(y) := (proj1x sr A)[proj0x/y], f sr (A ⇒ {α · B · β}) := ∀x[τS(A)] (x sr A ⇒ fx sr {α · B · β}), f sr (∀x {α(x) · A(x) · β(x)}) := ∀xD (fx sr {α(x) · A(x) · β(x)}), where for x : S → [τS(A)] × S we define x sr {α · A · β} := ∀πS ([α](π) ⇒ proj0(xπ) sr A ∧ [β](proj1(xπ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The following substitution lemma is easily proven by induction on formulas of SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For any term t of SL and s : [τS(A)] we have s sr A[t/x] = (s sr A)[t/x], where x is not free in s and on the right hand side we implicitly mean the natural interpretation of t in Sω (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='4 (Soundness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Suppose that Γ := Au1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , Aun n ⊢S {α · A · β} is provable in SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Then we can extract from the proof a term ∆, τS(Γ) ⊢ t : τS(A) of ST, where ∆ contains the free variables of Γ and {α · A · β} (typed with type D) and τS(Γ) := x1 : τS(A1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , xn : τS(An) for fresh variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , xn, such that the formula [t] sr {α · A · β} is provable in Sω from the assumptions (x1 sr A1)u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , (xn sr An)un for xi : [τS(Ai)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The theorem holds more generally for proofs in SL + ∆H + ∆S, now provably in Sω + ΛSω, if: for any axiom Γ ⊢H α in ∆H, the corresponding axiom [Γ](π) ⇒ [α](π) is added to ΛSω, for any axiom in ∆S there is a term t of ST+ΛS such that [t] realizes that axiom provably in Sω +ΛSω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Induction on the structure of derivations in SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' In all cases, we assume as global assumptions (x1 sr A1)u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , (xn sr An)un, and our aim is then to produce a term t such that if [α](π) holds for some state variable π, then a sr A and [β](π1) hold for ⟨a, π1⟩ := [t]π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For the axiom Γ ⊢S {α · A · α}, if Au ∈ Γ we define t := x for the corresponding variable x : τS(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Then [x]π := ⟨x, π⟩ for x sr A and [α](π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For Γ ⊢S {α · ⊤ · α} we define t := skip and the verification is even simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 12 (∧SI) Given terms s, t with [s] sr {α · A · β} and [t] sr {β · B · γ}, from [α](π) we can infer a sr A and [β](π1) for ⟨a, π1⟩ := [s]π, and from [β](π1) it follows that b sr B and [γ](π2) for ⟨b, π2⟩ := [t]π1, therefore we have shown that [s ◦ t] sr {α · A ∧ B · γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (∧SEi) If [t] sr {α · A ∧ B · β} then ⟨a, b⟩ sr A ∧ B and [β](π1) follow from [α](π), where ⟨a, b, π1⟩ := [t]π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' But then [p0t] sr {α · A · β} and [p1t] sr {α · B · β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (∨SIi) If [t] sr {α · A · β} and [α](π) holds, then a sr A and [β](π1) for ⟨a, π1⟩ := [t]π, and therefore (b = f ⇒ a sr A) ∧ (b = t ⇒ 0τS(B) sr B) for b := f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Thus [ι0t] sr A ∨ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' By an entirely analogous argument we can show that [ι1t] sr A ∨ B whenever [t] sr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (∨SE) Suppose that r, s(x) and t(y) are such that [r] sr {α · A ∨ B · β}, [s](x) sr {β · C · γ} assuming x sr A, and [t](y) sr {β · C · γ} assuming y sr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We claim that [elim r (λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='s) (λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t)] sr {α · C · γ} To prove this, first note that if [α](π), we have ⟨e, a, b⟩ sr A ∨ B and [β](π1) for ⟨e, a, b, π1⟩ := [r]π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' There are now two possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' If e = f then elim r (λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='s) (λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t)tπ = faπ2 for ⟨f, π2⟩ := [λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='s]π1 = ⟨λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [s](x), π1⟩ = (λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [s](x))aπ1 = [s](a)π1 But since [β](π1) holds and e = f also implies that a sr A, we have c sr C and [γ](π2) for ⟨c, π2⟩ := [s](a)π1, which proves the main claim in the case e = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' An analogous argument works for the case e = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (⇒S I) If t(x) is such that [t](x) sr {α · B · β} whenever x sr A, then by definition we have λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [t] sr A ⇒ {α · B · β} and therefore [λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t] sr {γ · A ⇒ {α · B · β} · γ} for any γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (⇒S E) Assume that [s] sr {β · A · γ} and [t] sr {α · A ⇒ {γ · B · δ} · β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' If [α](π) holds then defining ⟨f, π1⟩ := [t]π we have [β]π1 and x sr A ⇒ fx sr {γ · B · δ} Similarly, defining ⟨a, π2⟩ := [s]π1, it follows that [γ](π2) and a sr A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Finally, setting ⟨b, π3⟩ := faπ2 it follows that b sr B and [δ](π3), and we have therefore proven that [ts] sr {α · B · δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (⊥SE) If [t] sr {α · ⊥ · β} then from [α](π) we can infer a sr ⊥ and [β](π2) for ⟨a, π1⟩ := [t]π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' But a sr ⊥ = ⊥, and from ⊥ we can deduce anything, and in particular 0τS(A) sr A and [γ](π), from which it follows that [defaultτS(A)] sr {α · A · γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (∀SI) Suppose that t(x) is such that [t](y) sr {α[y/x] · A[y/x] · β[y/x]}, where y ≡ x or y is not free in {α · A · β}, and y is not free in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Then since y is not free in any of the assumptions xi sr Ai, we can deduce in Sω that ∀xD [t](x) sr {α · A · β} and therefore λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [t] sr ∀x {α · A · β}, and thus (just as for ⇒S I) we have [λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t] sr {γ · ∀x {α · A · β} · γ} for any γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 13 (∀SE) Suppose that [s] sr {α · ∀x {β · A · γ} · β[t/x]} and that [α](π) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Then f sr ∀x {β · A · γ} and [β][t/x](π1) for ⟨f, π⟩ := [s]π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Now, using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='8 we have [st]π = ftπ1 for the natural interpretation of t in Sω, since we can prove in Sω that ft sr {β[t/x] · A[t/x] · γ[t/x]} it follows that a sr A[t/x] and [γ][t/x](π2) for ⟨a, π2⟩ := ftπ1, and therefore we have shown that [st] sr {α · A[t/x] · γ[t/x]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (∃SI) If [s] sr {α · A[t/x] · β} and [α](π) then a sr A[t/x] and [β](π1) for ⟨a, π1⟩ := [s]π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3 we therefore have (a sr A)[t/x], and therefore ⟨t, a⟩ sr ∃x A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Observing (using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='8) that [t ◦ s]π = ⟨t, a, π1⟩, we have shown that [t ◦ s] sr {α · ∃x A · β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (∃SE) Suppose that s and t(x, z) are such that [s] sr {α · ∃x A · β} and z sr A[y/x] ⇒ [t](y, z) sr {β · C · γ} where y ≡ x or y is not free in A, and y is also not free in C, α, β, γ or Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3 that z sr A[y/x] = (z sr A)[y/x] = ⟨y, z⟩ sr ∃x A we therefore have ⟨y, z⟩ sr ∃x A ⇒ [t](y, z) sr {β · C · γ} Now, applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='9 to ∆, Γ, y : D, z : τS(A) ⊢ t : τS(C), we have [(λ∗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t)s]π = [t](e, a)π1 for ⟨e, a, π1⟩ := [s]π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Now, if [α](π) holds, then we have ⟨e, a⟩ sr ∃x A and [β](π1), and therefore since [t](e, a) sr {β · C · γ}, we have c sr C and [γ](π2) for ⟨c, π2⟩ = [t](e, a)π1 = [(λ∗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t)s]π, and thus we have shown that [(λ∗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t)s] sr {α · C · γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (cons) If α ⊢H β and γ ⊢H δ then by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3 both [α](π) ⇒ [β](π) and [γ](π) ⇒ [δ](π) are provable in Sω (respectively Sω + ΛSω for the general version of the theorem) for any π : S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' It is then easy to show that if [t] sr {β · A · γ} then we also have [t] sr {α · A · δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (cond) Suppose that [s] sr {α ∧ γ · A · δ} and [t] sr {β ∧ γ · A · δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We claim that [if α then s else t] sr {γ · A · δ} To prove this, suppose that [γ](π) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Since ⊢H α ∨ β then [α](π) ∨ [β](π) is provable in Sω, and so we consider two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Let {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , xn} be the free variables of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' If [α](π) holds, then [if α then s else t]π = χα ⟨x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , xn⟩ π ([s]π) ([t]π) = [s]π and since then [α](π) ∧ [γ](π) we have a sr A and [δ](π1) for ⟨a, π1⟩ := [s]π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' On the other hand, if [β](π) holds, then by an analogous argument we can show that a sr A and [δ](π1) for ⟨a, π1⟩ := [t]π = [if α then s else t]π, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The extension of the soundness theorem to SL + ∆H + ∆S is straightforward, as the soundness proof is modular and so any axioms along with their realizers can be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The first condition is needed so that Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3 (needed for the cons rule) continues to apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For the free variable condition that the free variables of t are contained in those of Γ, {α · A · β} and τS(Γ), if this were not the case, we could simply ground those variables with a canonical constant c : D and we would still have ˜t sr {α · A · β} for the resulting term ˜t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 14 Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='5 (Program extraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Suppose that the sentence ⊢S {α · ∀x {β · ∃y P(x, y) · γ(x)} · β} is provable in SL + ∆S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Then we can extract a closed realizing term t : D → D × C in ST + ΛS such that defining g : D → S → D × S by gxπ := ⟨a, π2⟩ for ⟨f, π1⟩ := [t]π and ⟨a, (), π2⟩ := fxπ1, we have ∀πS([α](π) ⇒ ∀xD (P(x, proj0(gxπ)) ∧ [γ](x)(proj1(gxπ)))) provably in Sω + ΛSω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1 Simplification and removal of unit types In presentations of modified realizability that use product types instead of type sequences, it is common to introduce the notion of a Harrop formula (a formula that does not contain disjunction or existential quantifi- cation in a positive position) and define realizability in a way that all Harrop formulas have unit realizability type, so that e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' τS(∀x (P ∧ Q)) = 1 for atomic predicates P and Q, rather than τS(∀x (P ∧ Q)) = D → 1×1 as for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We have avoided this simplification earlier on, as it would have added additional cases and bureau- cracy to our soundness theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' However, we can compensate retroactively for this choice by introducing equivalences on types that eliminate unit types, namely the closure under contexts of 1 × X ≃ 1 ≃ X × 1 (1 → X) ≃ X (X → 1) ≃ 1 along with corresponding equivalences on terms, also closed under contexts: t1×X ≃ proj1(t)X tX×1 ≃ proj0(t)X t1→X ≃ t() tX ≃ λx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t tX→1 ≃ () For example, in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='5 we would then have [t]π : (D → S → D × 1 × S) × S ≃ (D → S → D × S) × S and gxπ ≃ fxπ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For us, the equivalence relation ≃ will not play a formal role in the paper, but will be used to provide simplified descriptions of extracted programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='2 Examples of program extraction We now continue the short illustrative examples we outlined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='6 (Simple read-write).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' In Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='2 we considered a state where three actions were possible (writing to the state, performing a calculation, and reading the output from the state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We can formalise these three actions semantically in the metatheory Sω by including three constants in ΛSω, namely c1 : D → S → S, c2 : S → S and c3 : S → D, along with the characterising axioms: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' stored(x, c1xπ), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' stored(x, π) ⇒ solved(x, c2π), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' solved(x, π) ⇒ P(x, c3π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' While we are able to use these constants to form terms in Sω such as λπ, π1, x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' ⟨c1xπ, c2π1⟩, which could be viewed as non-sequential in the sense that we take two input states as arguments, we can force them to be applied in a sequential, call-by-value manner by adding three corresponding constants to our term calculus ST, namely including write : D → C, calc : C and read : D × C in ΛS, along with the embedding rules [write]π := ⟨λx, π′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' ⟨(), c1xπ′⟩, π⟩ ≃ ⟨c1, π⟩, [calc]π := ⟨(), c2π⟩ so that [calc] ≃ c2, 15 [read]π := ⟨c3π, (), π⟩ ≃ ⟨c3π, π⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' and then restricting out attention to terms of the form [t] for t ∈ ST + {write, calc, read}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We can then prove the following in Sω i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' that all axioms in ∆S can be realised: [write(x)] sr {α · ⊤ · stored(x)}, [calc] sr {stored(x) · ⊤ · solved(x)}, [read] sr {solved(x) · ∃y P(x, y) · ⊤}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' and thus Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='4 applies to SL + ∆H + ∆S for ∆H = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' In particular, we have [t] sr {β · ∀x {α · ∃y P(x, y) · ⊤} · β} for t := λx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' ((write(x) ∗ calc) ∗ read) where ∗ is sequential composition operator from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' A formal derivation of this term from the corresponding proof given in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='2 is as follows: x : D ⊢ write(x) : C x : D ⊢ calc : C ∧SI x : D ⊢ write(x) ◦ calc : C × C ∧SEL x : D ⊢ write(x) ∗ calc : C x : D ⊢ read : D × C ∧SI x : D ⊢ (write(x) ∗ calc) ◦ read : C × D × C ∧SEL x : D ⊢ (write(x) ∗ calc) ∗ read : D × C ∀SI ⊢ λx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' ((write(x) ∗ calc) ∗ read) : D → D × C Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='7 (Fixed-length array sorting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' In Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3 we considered a situation where we are allowed a single action on our state, namely to swap elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Analogously to the previous example, we can formalise this in our semantic environment Sω by adding to ΛSω constants cl,l′ : S → S for each pair l, l′ ∈ {1, 2, 3} along with the axiom [α](π) ⇒ [α[l ↔ l′]](cl,l′π) ranging over state formulas α of the form (2) and locations l, l′ ∈ {1, 2, 3} of SL, together with axioms corresponding to those of ∆H i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (π) ⇒ sorted(π) and [l ≤ l′ ∨ l′ ≤ l](π) Similarly, for each l, l′ ∈ {1, 2, 3} we add a term swapl,l′ : C to ΛS and define [swapl,l′]π := ⟨(), cl,l′π⟩ so that swapl,l′ sr {α · ⊤ · α[l ↔ l′]} A derivation of a closed term t : C of ST + {swapl,l′} such that [t] sr {⊤ · ⊤ · sorted} is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' In particular, we can prove in Sω that ∀πS sorted(proj1([t]π)), and so the term λπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' proj1([t]π) : S → S acts as a sorting program for arrays of length three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For an extracted term t corresponding to the proof given in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3, first we interpret D1 as ⊢ skip : C cons ⊢ skip : C 2↔3 ⊢ swap2,3 : C cons ⊢ swap2,3 : C cond[2≤3∨3≤2] ⊢ t1 := if (2 ≤ 3) then (skip) else (swap2,3) : C and define t1 := if (2 ≤ 3) then (skip) else (swap2,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Now D2 is interpreted as 1↔2 ⊢ swap1,2 : C D1 ⊢ t1 : C ∧SI ⊢ swap1,2 ◦ t1 : C × C ∧SEL ⊢ t2 := swap1,2 ∗ t1 : C 16 where we define t2 := swap1,2 ∗ t1 : C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Continuing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' D3 is interpreted as: D2 ⊢ t2 : C ⊢ skip : C cons ⊢ skip : C cond[2≤1∨1≤2] t3 := if (2 ≤ 1) then t2 else (skip) : C where t3 := if (2 ≤ 1) then t2 else (skip),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' and finally ⊢ skip : C 2↔3 ⊢ swap2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3 : C cond[2≤3∨3≤2] ⊢ if (2 ≤ 3) then (skip) else (swap2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3) : C D3 ⊢ t3 : C ∧SI ⊢ (if (2 ≤ 3) then (skip) else (swap2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3)) ◦ t3 : C × C ∧SEL ⊢ t := (if (2 ≤ 3) then (skip) else (swap2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3)) ∗ t3 : C 6 An extension to arithmetic We now present an extension of our framework to a stateful version of first-order intuitionistic arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' On the logic side, we will add not only a stateful induction rule, but also a Hoare-style while rule for iteration over the natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' On the computational side, these will be interpreted by stateful recursion in all finite types, along with a controlled while loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The addition of these constants will allow us to extract programs that are more interesting than those obtainable from proofs in pure predicate logic, and which can be clearly compared to well-known stateful algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' To exemplify this, we will present a formally synthesised version of insertion sort, and we stress that by further extending our framework with additional rules and terms, we would be able to extract an even richer variety of combined functional/stateful programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1 The system SA: First-order arithmetic with state Our system of stateful intuitionistic arithmetic SA builds on SL just as ordinary first-order Heyting arithmetic builds on first-order predicate logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' In both cases, we introduce a constant 0, a unary successor symbol succ, symbols for all primitive recursive functions, and our predicate symbols now include an equality relation =.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' In what follows we write x + 1 instead of succ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The axioms and rules of SA are, in turn, analogous to the additional axioms and rules we would require in ordinary first-order arithmetic: They include all axioms and rules of SL (based now on the language of SA), along with a collection of additional axioms and rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' These comprise not only basic axioms and rules for equality and the successor, and an induction rule (all now adapted to incorporate the state), but also a new while rule for stateful iteration, which now exploits our state and, as we will see, allows us to extract programs that contain while loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' These additional axioms and rules are outlined in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Our formulation of stateful arithmetic follows the same basic idea as the construction of stateful predicate logic, incorporating standard rules but keeping track of an ambient state in a call-by-value manner, and adding new rules that explicitly correspond to stateful constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' In particular, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1 clearly extends to SA, as the usual axioms and rules of arithmetic can be embedded into those of SA: Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For any formula A of HA and state formula α, define the main formula Aα of SA as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Then whenever Γ ⊢I A is provable in HA, we have that Γα, ∆ ⊢S {α · Aα · α} is provable in SA, where ∆ is arbitrary and Γα := (A1)u1 α , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , (An)un α for Γ := Au1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , Aun n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We can also derive a natural extensionality rule from our stateful equality rules, which assures us that whenever s = t in ordinary Heyting arithmetic, then we can replace s by t for stateful formulas: Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Suppose that ⊢I s = t is provable in HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Then from Γ ⊢S {α(s) · A(s) · β(s)} we can derive Γ ⊢S {α(t) · A(t) · β(t)} in SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 17 Figure 3: Additional axioms and rules of SA Axioms and rules for equality Γ ⊢S {α · t = t · α} Γ ⊢S {α · s = t · β} Γ ⊢S {α · t = s · β} Γ ⊢S {α · r = s · β} Γ ⊢S {β · s = t · γ} Γ ⊢S {α · r = t · γ} Γ ⊢S {α · s = t · β} Γ ⊢S {β · A(s) · γ(s)} ext Γ ⊢S {α · A(t) · γ(t)} Axioms and rules for arithmetical function symbols Γ ⊢S {α · succ(t) ̸= 0 · α} {α · succ(s) = succ(t) · β} {α · s = t · β} Γ ⊢S {α · l = r · α} where l = r ranges across defining equations for prim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' functions Induction rule Γ ⊢S {α · A(0) · β(0)} Γ, A(x) ⊢S {β(x) · A(x + 1) · β(x + 1)} ind Γ ⊢S {γ · ∀x {α · A(x) · β(x)} · γ} While rule (over natural numbers) A1 A2 A3 while Γ, A(x) ⊢S {α(x) · B · β} A1 := Γ, A(x + 1) ⊢S {γ(x + 1) ∧ α(x + 1) · A(x) · α(x)} A2 := Γ, A(x + 1) ⊢S {¬γ(x + 1) ∧ α(x + 1) · B · β} A3 := Γ, A(0) ⊢S {α(0) · B · β} for ind and while, x is not free in Γ, and for while it is not free in B or β Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1 for α := α(s) we have Γ ⊢S {α(s) · s = t · α(s)} and thus using the extensionality rule in SA we can derive Γ ⊢S {α(s) · s = t · α(s)} Γ ⊢S {α(s) · A(s) · β(s)} ext Γ ⊢S {α(s) · A(t) · β(t)} Since ⊢I t = s must also be provable in HA, another instance of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1 for α := α(t) along with the true axiom in SA gives us Γ ⊢S {α(t) · t = s · α(t)} Γ ⊢S {α(t) · ⊤ · α(t)} ext Γ ⊢S {α(t) · ⊤ · α(s)} Putting these together we obtain Γ ⊢S {α(t) · ⊤ · α(s)} Γ ⊢S {α(s) · A(t) · β(t)} ∧SI Γ ⊢S {α(t) · ⊤ ∧ A(t) · β(t)} ∧SEL Γ ⊢S {α(t) · A(t) · β(t)} which completes the derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='2 An extended term calculus STN In order to give derivations in SA a computation interpretation, we need to extend our term calculus ST to include a recursor (for induction) and a controlled while loop (for the while rule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The remaining new axioms and rules of SA are dealt with in a straightforward manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' To be precise: the theory STN is defined to be the instance of ST for the case of arithmetic, with function symbols for zero, successor and all primitive recursive functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Accordingly, we rename the base type D to Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' In addition to the terms of ST, we add terms rec e e and whilee γ[z]e e e to our grammar, where γ[z] ranges over state formulas of SL with a specified free variable z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The typing rules for these new terms are Γ ⊢ s : X Γ ⊢ t : Nat → X → X Γ ⊢ rec s t : Nat → X for the recursor, while for the while loop we have Γ ⊢ r : Nat → X → X Γ ⊢ s : Nat → X → Y Γ ⊢ t : X → Y Γ ⊢ u : Nat Γ ⊢ whileu γ[z] r s t : X → Y under the additional variable condition that z /∈ Γ, but x : Nat ∈ Γ for all free variables of γ[z] outside of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Note that we do not consider z a free variable of whilea γ[z] r s t, but rather a placeholder for the loop condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' In order to give the appropriate semantics to our terms, we must add to our metatheory Sω axioms and rules for arithmetic in all finite types, including the ability to define functions of arbitrary type via recursion over the natural numbers, along the lines of E-HAω [30] (though as before the precise details are not important).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We then define: [rec s t]π := ⟨Rf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' π1⟩ for ⟨f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' π1⟩ := [t]π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' where Rf0π := [s]π Rf(n + 1)π := gaπ′ 2 for ⟨a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' π′ 1⟩ := Rfnπ′ and ⟨g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' π′ 2⟩ := fnπ′ 1 (3) [whileu γ[z] r s t]π := ⟨Lf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='hm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' π4⟩ where ⟨f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' π1⟩ := [r]π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' ⟨g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' π2⟩ := [s]π1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' ⟨h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' π3⟩ := [t]π2 and ⟨m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' π4⟩ := [u]π3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' where Lf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='h0yπ′ := hyπ′ Lf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='h(n + 1)yπ′ := � Lf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='hny′π2 for ⟨a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' π′ 1⟩ := fnπ′ and ⟨y′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' π′ 2⟩ := ayπ′ 1 if [γ][n + 1](π′) byπ′ 1 for ⟨b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' π′ 1⟩ := gnπ′ if ¬[γ][n + 1](π′) (4) where in the case distinctions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' we would technically speaking need to use the characteristic function χγ⟨x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , xk⟩ for γ, with n substituted for the special free variable z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3 The soundness theorem for arithmetic We now need to show that the soundness proof for stateful predicate logic also holds in the extension to arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The statement of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='4 remains valid if we replace SL by SA and ST by STN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We need to extend the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3 to show that the additional axioms and rules as in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1 can be realized by a term of the form [t] for t in STN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For the non-extensionality equality and arithmetic axioms this is straightforward due to the fact that these are also true in Sω: For instance, given a realizer [s] sr {α · u = v · β} and [t] sr {β · v = w · γ}, we have that [s ◦ t] sr {α · u = v ∧ v = w · γ}, and since from u = v ∧ v = w we can infer u = w in Sω, it follows that [p1(s ◦ t)] sr {α · u = w · γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The other axioms and rules are even simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 19 (ext) Extensionality is similarly simple: If [s] sr {α · u = v · β} and [t] sr {β · A(u) · γ(u)}, then [α](π) implies that u = v and [β](π1) for ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , π1⟩ := [s]π, and therefore a sr A(u) and [γ](u)(π2) for ⟨a, π2⟩ := [t]π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Now applying extensionality in Sω to the formula T (x) := a sr A(x) ∧ [γ](x)(π2), from u = v we have a sr A(v) and [γ](v)(π2), and thus [s ◦ t] sr {α · u = v ∧ A(v) · γ(v)} and therefore [p2(s ◦ t)] sr {α · A(v) · γ(v)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (rec) Suppose that s and t(x, y) are such that [s] sr {α · A(0) · β(0)} and [t](x, y) sr {β(x) · A(x + 1) · β(x + 1)} assuming y sr A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We show that [rec s λx, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t(x, y)] sr {γ · ∀x {α · A(x) · β(x)} · γ} for any γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Since [rec s λx, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t(x, y)]π = ⟨Rf, π⟩ for f := λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t(x, y)] and Rf as in (3), it suffices to show that for any n : Nat we have Rfn sr {α · A(n) · β(n)} We prove this by induction: For the base case, we have Rf0 = [s] and the claim holds by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For the induction step, let us assume that [α](π′) holds, and so by the induction hypothesis we have a sr A(n) and [β(n)](π′ 1) for ⟨a, π′ 1⟩ := Rfn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Since fnπ′ 1 = ⟨g, π′ 1⟩ for g := λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [t](n, y), we have that Rf(n+1)π′ = [t](n, a)π′ 1, and since by the property of [t] we then have b sr A(n + 1) and [β(n+1)](π′ 2) for ⟨b, π′ 2⟩ := [t](n, a)π′ 1, we have shown that Rf(n + 1) sr {α · A(n + 1) · β(n + 1)}, which completes the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (while) We suppose that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [r](x, y) sr {γ(x + 1) ∧ α(x + 1) · A(x) · α(x)} assuming that y sr A(x + 1), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [s](x, y) sr {¬γ(x + 1) ∧ α(x + 1) · B · β} assuming that y sr A(x + 1), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [t](y) sr {α(0) · B · β} assuming that y sr A(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Our aim is to show that [(whilex γ (λx′, y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='r) (λx′, y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='s) (λy′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t))y] sr {α(x) · B · β} for any x, y ∈ Nat with y sr A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We observe, unwinding the definition, that [(whilex γ (λx′, y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='r) (λx′, y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='s) (λy′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t))y]π = Lf,g,hxyπ for f := λx′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [λy′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='r(x′, y′)], g := λx′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [λy′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='s(x′, y′)], h := λy′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [t](y′) and Lf,g,h as defined in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We now show by induction on n that if y sr A(n) then Lf,g,hny sr {α(n) · B · β} and then the result follows by setting n := x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The base case is straightforward since Lf,g,y0y = [t](y) and the claim follows by definition of [t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For the induction step, suppose that y sr A(n + 1) and [α(n + 1)](π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' There are two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' If ¬[γ](n + 1)(π) we have Lf,g,h(n + 1)yπ = [s](n, y)π and the result holds by the property of [s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' On the other hand, if [γ](n + 1)(π) then Lf,g,h(n + 1)yπ = Lf,g,hny′π′ for ⟨y′, π′⟩ := [r](n, y)π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' But by the property of [r] we have y′ sr A(n) and [α(n)](π′), and therefore by the induction hypothesis we have b sr B and [β](π′′) for ⟨b, π′′⟩ := Lf,g,hny′π′ = Lf,g,h(n + 1)yπ, and so the result is proven for n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' This covers all the additional axioms and rules of SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='4 Worked example: Insertion sort We now illustrate our extended system by synthesising a list sorting program that, intuitively, forms an implementation of the insertion sort algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Here our state will represent the structure that is to be sorted, and continuing the spirit of generality that we have adhered to throughout, we characterise this structure through a number of abstract axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Instantiating the state as, say, an array of natural numbers, would provide a model for our theory, but our sorting algorithm can be extracted on the more abstract level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Crucially, the proof involves both loop iteration and induction, and the corresponding program combines an imperative while loop with a functional recursor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We begin by axiomatising our state, just as in previous examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' An intuition here is that states represent an infinite array of elements a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' possessing some total order structure ≤, and we seek to extract a program that, for any input n, sorts the first n elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We use this informal semantics throughout to indicate the intended meaning of our axioms, but stress that none of this plays a formal role in the proof or resulting computational interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We introduce three state predicates to SA, with the intuition indicated in each case: sort(N) Sorted: The first N + 1 elements of the array i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , aN] are sorted psort(n, N) Partially sorted with respect to an: if n < N then the list [a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , an−1, an+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , aN] is sorted and an ≤ an+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For the base cases, if n = N then the list [a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , aN−1] is sorted, and if n > N then the list [a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , aN] is sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' comm(n) Comparison: true if an ≤ an−1, and always true if n = 0 We formalise this intuition by adding the following state independent axioms to ∆H: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Γ, sort(N) ⊢H psort(N + 1, N + 1) If the first N +1 elements are sorted, then they are also partially sorted with respect to the next element aN+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Γ, ¬comm(n), psort(n, N) ⊢H sort(N) If [a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , an−1, an+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , aN] is sorted, an ≤ an+1, but also an−1 ≤ an, then the entire segment [a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , aN] must be sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Γ, psort(0, N) ⊢H sort(N) If [a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , aN] is sorted and a0 ≤ a1, then [a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , aN] is sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Γ ⊢H sort(0) The singleton array [a0] is defined to be sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We complete the axiomatisation by adding a single state-sensitive axiom to ∆S: 5 Γ ⊢S {comm(n + 1) ∧ psort(n + 1, N) · ⊤ · psort(n, N)} If [a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , an, an+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , aN] is sorted and an+1 ≤ an+2, but an+1 ≤ an, then we can modify the state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' swapping an and an+1 by setting ˜an := an+1 and ˜an+1 := an) so that [a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , an−1, ˜an+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' , aN] is sorted and ˜an ≤ ˜an+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The edge cases for n ≥ N are interpreted in a more straightforward way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' In order to give a realizing term to this axiom, we representing element swapping semantically by adding a constant c : Nat → S → S to our metatheory Sω, which satisfies comm(n + 1, π) ∧ psort(n + 1, N, π) ⇒ psort(n, N, cnπ) 21 and a corresponding term swap : Nat → C to our term calculus, along with the embedding [swap]π := ⟨λn, π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='⟨(), cnπ⟩, π⟩ ≃ ⟨c, π⟩ so that we can prove [swap n] sr {comm(n + 1) ∧ psort(n + 1, N) · ⊤ · psort(n, N)} With this in place, we can now prove in SA that the first N elements of the state can be sorted, and extract a corresponding realizing term in STN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1 Proof of ⊢S {γ · ∀N {α · ⊤ · sort(N)} · γ} in SA The core of our proof begins with an instance of the while rule parametrised by N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' with Γ := ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' A(n) := ⊤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' α(n) := psort(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' N + 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' β := sort(N + 1) and γ(n) := comm(n): D1 D2 D3 while ⊤ ⊢S {psort(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' N + 1) · ⊤ · sort(N + 1)} ∀SI ⊤ ⊢S {psort(N + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' N + 1) · ∀n {psort(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' N + 1) · ⊤ · sort(N + 1)} · psort(N + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' N + 1)} ∀SE ⊤ ⊢S {psort(N + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' N + 1) · ⊤ · sort(N + 1)} cons ⊤ ⊢S {sort(N) · ⊤ · sort(N + 1)} where the final composition inference makes use of the first state independent axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Here D1 represents an instance of the state sensitive axiom ⊤ ⊢S {comm(n + 1) ∧ psort(n + 1, N + 1) · ⊤ · psort(n, N + 1)} and D2 represents the derivation ⊤ ⊢S {sort(N + 1) · ⊤ · sort(N + 1)} cons ⊤ ⊢S {¬comm(n + 1) ∧ psort(n + 1, N + 1) · ⊤ · sort(N + 1)} where composition makes use of the second state independent axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Finally D3 is ⊤ ⊢S {sort(N + 1) · ⊤ · sort(N + 1)} cons ⊤ ⊢S {psort(0, N + 1) · ⊤ · sort(N + 1)} this time making use of the third state independent axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Finally we can prove that all lists can be sorted with an outer induction as follows: ⊢S {α · ⊤ · α} cons ⊢S {α · ⊤ · sort(0)} D ⊤ ⊢S {sort(N) · ⊤ · sort(N + 1)} ind ⊢S {γ · ∀N {α · ⊤ · sort(N)} · γ} where α is an arbitrary state predicate, the instance of cons uses the fourth state independent axiom, and D represents the derivation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='2 Program extraction We now extract a program that corresponds to the above proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' First of all, we note that the three premises of our while rule are realised by swap n, skip and skip respectively, and so our derivation D corresponds to the following program: y : C ⊢ swap n : C y : C ⊢ skip : C y : C ⊢ skip : C while y : C ⊢ t(n)y : C ∀SI y : C ⊢ λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t(n)y : Nat → C ∀SE y : C ⊢ (λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t(n)y)(N + 1) : C cons y : C ⊢ (λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t(n)y)(N + 1) : C 22 where t(n) := whilen comm[z] (λx, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (swap x)) (λx, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='skip) (λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='skip) ≃ whilen comm[z] (λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (swap x)) (skip) (skip) Then our final induction generates the following program: ⊢ skip : C y : C ⊢ (λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t(n)y)(N + 1) : C ind ⊢ rec (skip) (λx, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' ((λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t(n)y)(x + 1))) : Nat → C Thus our list sorting program is rec (skip) (λx, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' ((λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='t(n)y)(x + 1))) ≃ rec (skip) (λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='((λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (whilen comm[z] (λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' (swap x)) (skip) (skip)()))(x + 1))) which is essentially an implementation of the insertion sort algorithm, with an outer recursion that sorts initial segments of the list in turn, and an inner loop that inserts new elements into the appropriate place in the current sorted list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 7 Directions for future work In this paper we have presented the central ideas behind a new method for extracting stateful programs from proofs, which include an extension of ordinary first-order logic with Hoare triples, a corresponding realizability interpretation, and a soundness theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We emphasise once again that our intention has been to offer an alternative approach to connecting proofs with stateful programs, one that seeks to complement rather than improve existing work by embracing simplicity and abstraction, and which might be well suited to a range of applications in proof theory or computability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' In this spirit, we conclude with a very informal outline of a series interesting directions in which we anticipate that our framework could be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='1 Further extensions and program synthesis While our main results have been presented in the neutral setting of first-order predicate logic, it would be straightforward to extend SL to richer logics with more complex data structures and a imperative commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Already, the addition of recursion and loops over natural numbers in Section 6 has allowed us to synthesise a standard in-place sorting algorithm using our abstract axiomatisation of an ordered state, in a similar spirit to [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' However, further extensions are naturally possible, including the addition of general fixpoint operators and non-controlled while loops, which would then require a Sω to be replaced by a domain theoretic semantics that allows for partiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Looking a step further ahead, by implementing all of this in a proof assistant, we would have at our disposal a new technique for synthesising correct-by-construction imperative programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' While we do not suggest that this pipeline would directly compete with existing techniques for verifying imperative programs, it could be well suited to synthesising and reasoning about programs in very specific domains, where we are interested in algorithms for which interactions with the state have a restricted form that could be suitably axiomatised within our logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For example, a more detailed axiomatisation our state as an ordered array along the lines of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='4, with a “swap” operation and a few other ways of interacting with the state, might give rise to an interesting theory of in-place sort algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Stateful algorithms on other data structures, such as graphs, could presumably also be formalised within our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='2 Bar recursion and the semantics of extracted programs Two of the main starting points for this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' the monadic realizability of Birolo [7] and the author’s own Dialectica interpretation with state [24],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' address the broader problem of trying to understand the operational semantics of programs extracted from proofs as stateful procedures (the origins and development of this general idea,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' from Hilbert’s epsilon calculus onwards,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' is brilliantly elucidated in Chapter 1 of Aschieri’s 23 thesis [2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' who then sets out his own realizability interpretation based on learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' A number of case studies by the author and others [20, 21, 25, 26] have demonstrated that while terms extracted from nontrivial proofs can be extremely complex, they are often much easier to understand if one focuses on the way they interact with the mathematical environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For example, in understanding a program extracted from a proof using Ramsey’s theorem for pairs [20], it could be illuminating to study the trace of the program as it queries a colouring at particular pairs, as this can lead to a simpler characterisation of the algorithm ultimately being implemented by the term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' While the aforementioned analysis of programs has always been done in an informal way, our stateful realizability interpretation would in theory allow us to extract programs which store this trace formally in the state, where our abstract characterisation of state would allow us to implement it in whichever way is helpful in a given setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For example, in the case of the Bolzano-Weierstrass theorem [21], our state might record information of the form xn ∈ I, collecting information about the location of sequence elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For applications in algebra [26], one might instead store information about a particular maximal ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The aforementioned theorems are typically proven using some form of choice or comprehension, and that in itself leads to the interesting prospect of introducing both stateful recursors and while-loops that are computationally equivalent to variants of bar recursion [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' In [23], several bar recursive programs that arise from giving a computational interpretation to arithmetical comprehension principles are formulated as simple while loops, and these could in principle be incorporated into our system with new controlled Hoare rules in the style of update recursion [5], that replace the conditions n < N and n ≥ N in the Ai above with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' n ∈ dom(f) and n /∈ dom(f), where f is some partial approximation to a comprehension function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' An exploration of such while-loops from the perspective of higher-order computability theory might well be of interest in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='3 A logic for probabilistic lambda calculi Probabilistic functional languages are a major topic of research at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' While work in this direction dates back to the late 1970s [16, 28] where it typically had a semantic flavour, a more recent theme [9, 11, 12] has been to study simple extensions of the lambda calculus with nondeterministic choice operators ⊕, where s⊕t evaluates nondeterministically (or probabilistically) to either s or t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' While such calculi have been extensively studied, corresponding logics that map under some proof interpretation to probabilistic programs are far more rare (although there is some recent work in this direction e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We conjecture that our framework offers a bridge between logic and probabilistic computation through incorporating probabilistic disjunctions into our logic SL and taking states to be streams of outcomes of probabilistic events together with a current ‘counter’ that increases each time an event occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' In a simple setting where only two outcomes are possible with equal probability, we can axiomatise this within SL by adding zero and successor functions (allowing us to create numerals n), along with a unary state predicate count(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We can then model probabilistic events by adding the appropriate axioms to ∆S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Suppose, for example, we add two predicate constants H(x) and T (x) (for heads and tails), along with constants c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' representing coins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Then flipping a coin would be represented by the axiom schema Γ ⊢S {count(n) · H(ci) ∨ T (ci) · count(n + 1)} where n ranges over numerals and ci over coin constants, the counter indicating that a probabilistic event has occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' The act of reading a probability from the state could be interpreted semantically by introducing a constant ω : S → Bool × S to Sω, with the axiom count(n, π) ⇒ (e = f ⇒ H(ci)) ∧ (e = t ⇒ T (ci)) ∧ count(n + 1, π1) for ⟨e, π1⟩ := ωπ (alternatively, we could simply define S := Nat × (Nat → Bool) for a type of Nat natural numbers, and define ω⟨n, a⟩ := ⟨a(n), ⟨n + 1, a⟩⟩ and count(n, ⟨m, a⟩) := m =Nat n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' A probabilistic choice operator ⊕ can then be added to the language of ST,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' along with the typing rule Γ ⊢ s ⊕ t : X + Y for Γ ⊢ s : X and Γ ⊢ t : Y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' and the interpretation [s ⊕ t]π := case e ([ι0s]π1) ([ι1t]π1) where ⟨e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' π1⟩ := ωπ 24 In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' defining flip := skip ⊕ skip : C + C we would have [flip] sr {count(n) · H(ci) ∨ T (ci) · count(n + 1)} although we stress that the operator ⊕ and would allow for much more complex probabilistic disjunctions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' potentially involving additional computational content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Our soundness theorem, extended to these new probabilistic axioms and terms, would then facilitate the extraction of probabilistic programs from proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' For instance, including a winner predicate W(x), two player constant symbols p1, p2, and adding axioms H(c1), H(c2) ⊢S {α · W(p1) · α};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' T (c1), T (c2) ⊢S {α · W(p1) · α};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' H(c1), T (c2) ⊢S {α · W(p2) · α} and T (c1), H(c2) ⊢S {α · W(p2) · α} for any α, we could prove ⊢S {count(n) · ∃x W(x) · count(n + 2)} expressing the fact that a winner can be determined after two flips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We can then extract a corresponding probabilistic term for realizing this statement, which would be isomorphic to the expected program that queries the state twice in order to determine the outcome of those flips, and returns either p1 or p2 as a realizer for ∃x W(x) depending on the content of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' Of course, the details here need to be worked through carefully in order to properly substantiate the claim that our framework could be used to extract probabilistic programs in a natural and meaningful way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' At the very least, it is likely that further additions to SL along with a more intricate state would be needed to incorporate more interesting probabilistic events, such as annotated disjunctions along the lines of [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf'} +page_content=' We leave such matters to future work.' metadata={'source': 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+++ b/29E4T4oBgHgl3EQfawyS/content/2301.05067v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6cc27b663bf3916f6fa3d575e24f5586378492a8574983cfe7e4b367a7c369e7 +size 6842265 diff --git a/29FQT4oBgHgl3EQfGTVE/content/tmp_files/2301.13244v1.pdf.txt b/29FQT4oBgHgl3EQfGTVE/content/tmp_files/2301.13244v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bab08137d8c4d87c07844ce11479816fbc42ce8d --- /dev/null +++ b/29FQT4oBgHgl3EQfGTVE/content/tmp_files/2301.13244v1.pdf.txt @@ -0,0 +1,991 @@ +Mono-STAR: Mono-camera Scene-level Tracking and Reconstruction +Haonan Chang1, Dhruv Metha Ramesh1, Shijie Geng1, Yuqiu Gan, Abdeslam Boularias1 +Abstract— We present Mono-STAR, the first real-time 3D +reconstruction system that simultaneously supports semantic +fusion, fast motion tracking, non-rigid object deformation, and +topological change under a unified framework. The proposed +system +solves +a +new +optimization +problem +incorporating +optical-flow-based 2D constraints to deal with fast motion +and a novel semantic-aware deformation graph (SAD-graph) +for handling topology change. We test the proposed system +under various challenging scenes and demonstrate that it +significantly +outperforms +existing +state-of-the-art +methods. +Supplementary material, including videos, can be found at +https://github.com/changhaonan/Mono-STAR-demo. +I. INTRODUCTION +Real-time perception is a crucial component of modern +robotic manipulation systems. Recently, You Demonstrate +Only Once [1] has demonstrated that given the geometry +model and 6D-pose trajectory of a manipulated object during +an expert demonstration, a robot can quickly learn complex +and contact-rich manipulation skills. Such progress shows +the importance of geometric 3D reconstruction and tracking +systems for robotic manipulation. +However, a perception system that can perform both track- +ing and reconstruction simultaneously is notoriously difficult +to build because reconstruction and tracking inherently de- +pend on each other. For example, tracking algorithms usually +require geometry models, while dynamic scene reconstruc- +tion relies on accurate tracking for producing those geometry +models. Scene-level Tracking and Reconstruction (STAR) [2] +refers to a category of perception systems that generate both +the geometry and the pose of every visible object in a scene. +This problem is related to the multiple-instance dynamic +SLAM problem, where all movable objects in the scene are +assumed to be rigid so that the problem can be decomposed +into multiple dense-SLAM sub-problems. This approach was +proposed in Co-Fusion [3] and MaskFusion [4], where a +semantic neural network was employed first to decompose +the scene into multiple objects and then deal with each +object individually. This approach requires every object in +the scene to be rigid or quasi-rigid. The same problem was +investigated in MidFusion [5], where an octree was used to +improve reconstruction and tracking. However, these systems +are limited to scenes of rigid objects with slow motions. +Instead of dealing with each object individually based +on their semantic labels, one can also reconstruct all the +objects in the scene as one large non-rigid object and segment +them ulteriorly. This approach was however very challenging +1 Authors are with the Department of Computer Science, Rutgers +University, 08854 New Brunswick, USA. This work is supported by NSF +awards 1734492, 1846043, and 2132972. +to apply until the introduction of the first real-time non- +rigid reconstruction DynamicFusion [6], where the non- +rigid reconstruction problem was decomposed into two sub- +problems, (1) building the geometry at the initial frame, and +(2) computing the deformation using an embedded defor- +mation graph, namely ED-graph. This paradigm was also +followed in OcclusionFusion [7]. Inspired by these previous +efforts, a solution to the general STAR problem was recently +proposed in STAR-no-prior [2]. In contrast to SLAM-based +methods, STAR-no-prior reverses the order of segmentation +and reconstruction. The entire scene is first reconstructed and +then segmented into different objects based on topology. By +doing so, STAR-no-prior outperforms previous state-of-the- +art methods such as [4] and MidFusion [5]. +However, a major limitation of STAR-no-prior is its re- +liance on a system of multiple cameras surrounding the +scene, making it impractical for a mobile robot. To address +this shortcoming, we propose Mono-STAR, a mono-camera +STAR solution. Switching from a multi-camera system to +a mono-camera setting requires solving several non-trivial +problems. Notably, STAR-no-prior relies on the multi-camera +system to overcome the plane-based-ICP constraint that it +inherits from DynamicFusion, which supports tracking of +only slow motion along the camera view. The use of multiple +cameras can guarantee that any motion has at least one non- +zero projection to a camera view. However, the mono-camera +setting does not have such a guarantee and therefore requires +a new solution. Occlusion Fusion [7] adds a 2D constraint +using optical-flow (RAFT [8] or GMA [9]) to deal with fast +motions. Inspired by this, we propose a new 2D loss to track +motions that are perpendicular to the camera view, which not +only stabilizes tracking performance under a single view but +also improves our system’s ability to handle fast motion. +Furthermore, STAR-no-prior does not take advantage of +semantic labels. We, therefore, combine the semantic infor- +mation with the embedded deformation graph mechanism +and propose a Semantic-aware Adaptive Deformation graph, +SAD-graph, which is an extension of ED-graph. With just lit- +tle extra computation, SAD-graph can easily handle topology +changes across distinct semantic classes and assign different +levels of rigidness for each type of object. To the best of our +knowledge, Mono-STAR is the first single-view real-time 3D +reconstruction system that can simultaneously handle seman- +tic fusion, fast motion tracking, non-rigid object deformation, +and topological change under one unified framework. +II. RELATED WORKS +Simultaneous Tracking and Reconstruction. Simulta- +neous 6D tracking and 3D reconstruction was typically +arXiv:2301.13244v1 [cs.RO] 30 Jan 2023 + +Method +Semantic +Fast +Non-rigid +Topology +Single +motion +objects +change +view +SLAM++ + + + + + +DynamicFusion [6] + + + + + +Volume Deform + + + + + +SurfelWarp [10] + + + + + +TCAFusion [11] + + + + + +Co-fusion [3] + + + + + +Fusion4D [12] + + + + + +Motion2Fusion [13] + + + + + +Functon4D [14] + + + + + +MaskFusion [4] + + + + + +RigidFusion [15] + + + + + +MidFusion [5] + + + + + +OcclusionFusion [7] + + + + + +STAR-no-prior [2] + + + + + +Mono-STAR + + + + + +TABLE I: Taxonomy of the-state-of-art scene-level fusion systems. +regarded in previous works as a multiple-instance dynamic +SLAM problem. Many works such as Co-fusion [3], Mask- +Fusion [4], and RigidFusion [15] proposed to divide the +scene into multiple rigid objects and track each object +individually. More recently, STAR-no-prior [2] formalized +the STAR problem as a scene-level non-rigid reconstruction +problem. Our mono-camera system eliminates the multi- +camera requirement of STAR-no-prior by adding a new +optical-flow-based 2D constraint and a novel semantic-aware +adaptive deformation graph. +Dynamic Scene Reconstruction. Dynamic scene recon- +struction [16], [17] is the problem of reconstructing the +geometry and recording the deformation of a scene with +moving objects. DynamicFusion [6] was the first real-time +GPU-based solution for solving this problem. It adopts a +TSDF-based geometry as the canonical model and an embed- +deformation graph (ED-graph) to describe the deformation +of the whole scene. A drawback of this method is that the +combination of TSDF and ED-graph cannot handle topology +changes. Many recent techniques such as Fusion4D [12], +Motion2Fusion [13], [11], [18] have attempted to address this +problem. However, these methods require significantly more +computation or rely on expensive sensors. SurfelWarp [10] +demonstrated that a Surfel-based representation can be used +to tackle topology changes. Therefore, our proposed system +also adopts a Surfel-based representation. +III. PROBLEM FORMULATION AND BACKGROUND +A. Problem formulation +Given a sequence of RGB-D images of a given dynamic +scene taken from a single fixed camera, we consider the +problem of simultaneous tracking and reconstruction of all +the objects visible in the scene. The number of objects is +unknown. The objects can be non-rigid. Measurement, Mt +can defined as set of measurement surfels mi at time-step +t, generated from the RGB-D input. mi = (vi,ni,ci), where +v,n,c are 3D coordinates, normal and color respectively. +The proposed system returns at each time-step t a Surfel- +based geometry St (the reconstructed scene) for the entire +scene and its corresponding deformation graph Gt. Surfel- +based geometry St is a set of surfels si. si = (vi,ni,ci,ri,lsi), +where vi,ni,ci,ri,lsi are respectively the 3D coordinates, +normal, color, radius and semantic label of surfel si ∈ St. We +assume that there is a maximum of H pre-defined different +semantic categories {1,2,...,H}. If a surfel does not belong +to any pre-defined category, it will be labeled as H +1 (i.e., +unrecognized). Deformation graph Gt is defined by a set of +nodes {gi}. Each node gi has a semantic label lgi, and is +connected to its nearest-neighbor nodes, denoted as NG(gi), +in the 3D space. Deformation graph Gt is associated with +a warp field Wt, defined as W = {[pi ∈ R3,δi ∈ R+,Ti ∈ +SE(3)]}, wherein i is the index of a node in Gt, pi is the 3D +point that corresponds to node gi, δi is the node’s radius of +influence, and Ti is the 6D transformation defined on node +gi. Ti is represented by a dual quaternion qi for smooth +interpolation [19]. Warp field W describes the deformation +between two consecutive time steps. For each surfel s = +(v,n,c,r,l) ∈ S, we compute its 6D transformation +¯W(s) +based on warp field W, +¯W(s) = normalize( ∑ +k∈NG(s) +w(v, pk)qk), +(1) +wherein NG(s) denotes the neighbors nodes of surfel s, +w(s) is an interpolation parameter, defined as w(s) = +exp +� +∥v− pk∥2 +2 /(2δ 2 +k ) +� +, and v is the 3D position of surfel +s. The local transformation ¯W(s) is then used to describe the +deformation of surfel s as follows: +vwarp = ¯W(s)v +nwarp = rotation +� ¯W(s) +� +n. +(2) +Here, v,n are the vertex and normal of s before warping, and +vwarp,nwarp are the vertex and normal after the deformation. +Symbol +Meaning +Definition +Mt +Measurement at time t. +III-A, IV-A.1 +St−1 +Surfel geometry from t-1. +III-A +Ra +t−1 +2D maps rendered from St−1. +IV-B.1 +Swarp +t−1 +Warped geometry after non-rigid alignment. IV-B.4 +Rg +t−1 +2D maps rendered from Swarp +t−1 . +IV-B.1 +TABLE II: Notation sheet. +IV. PROPOSED APPROACH +An overview of the proposed method is shown in Fig. 1. +Mono-STAR uses two parallel threads, a measurement thread, +and a geometry thread. The first thread is responsible for +loading measurements Mt and generating semantic labels +Lm +t . The geometry thread uses this measurement Mt and the +alignment rendering Ra +t−1 to compute an optical-flow OFt. +Then, Mt,Ra +t−1,OFt are given to the optimization module that +then computes the non-rigid deformation Wt. After the non- +rigid alignment, previous geometry St−1 is warped to Swarp +t−1 , +and the geometry rendering Rg +t−1 is generated. Finally, Rg +t−1, +deformation Wt, warped geometry Swarp +t−1 , and semantic labels +Lm +t are combined to generate the latest geometry St. +Noticeably, loading Mt takes less time than updating +geometry St−1 and rendering Ra +t−1. Semantic segmentation is +also faster than the combined process of generating optical- +flow OFt and the non-rigid alignment. Thus, the geometry +thread fully hides the latency of the measurement thread. + +Fig. 1: Overview of the proposed system. The system runs in two parallel threads, one for measurement and one for geometry. In each +time-step t, the measurement thread loads a measurement Mt from images or a camera buffer. Then, a segmentation network generates a +set of semantic labels Lmt . Once the measurement is loaded on the GPU memory, Mt and previous alignment rendering Ra +t−1 are fed into +an optical-flow network to generate the optical-flow OFt from previous geometry St−1 to measurement Mt. Optical-flow OFt, geometry +rendering Rt and measurement Mt are used to compute warp-field Wt with non-rigid alignment. After the alignment, previous geometry +St−1 will be warped to Swarp +t−1 . The fusion rendering map Rg +t−1 is then rendered from Swarp +t−1 . Rg +t−1, Swarp +t−1 +and semantic labels Lmt are used +to generate the updated geometry St, deformation graph Gt and the surfel semantic label Lst . +A. Measurement Thread +1) Measurement: We use one Intel RealSense-415 camera +to collect RGB-D images. Depth images are denoised with a +Gaussian filter. The maximum frame rate for this module is +limited to 20 fps to coordinate with other modules. We use +a double-buffer strategy to hide latency. Specifically, we use +two buffers B0 and B1 to store measurements. When B0 is +used by other threads, B1 can read images simultaneously. +The filtered images are used to construct three maps, V m +t , +Nm +t , Cm +t , storing 3D coordinates vm +i , normal nm +i , and color cm +i , +respectively, for surfel mi of each pixel in the measurement. +2) Segmentation: The segmentation module receives the +color map Cm +t +and returns a semantic label map Lm +t +of H +pre-defined semantic classes. Here, we use two different +segmentation models, a transformer-based Segmenter Mask +[20], and a more traditional MaskRCNN [21]. The two +models are pre-trained on two different datasets, ADE20K +[22], [23] and COCO-Stuff [24] respectively. We do not +further train these models on any other dataset. We select +which one to use based on the types of objects in the scene. +B. Geometry Thread +1) Geometry Rendering: The input of the geometry ren- +dering pipeline is a geometry S, and the output is the +rasterized rendering R for geometry S from the current +camera view. The rendering process to generate R follows +the classical point cloud rasterizing process [25], where every +surfel is projected to its nearest pixel position on the camera +plane based on its 3D coordinates. Each rendering map R +is composed of four 2D maps. Rt = {Ct,Vt,Nt,It}, where +Vt,Nt,Ct,It are respectively the vertex map, the normal map, +the color map and the index map. These maps store the 3D +coordinates vi, the normal ni, the color ci and the surfel index +i of the projected surfel si at each pixel. +At each time-step, the geometry rendering pipeline is +called twice; once to generate Ra +t−1 from previous geometry +St−1 for non-rigid alignment, and once to get Rg +t−1 with +warped geometry Swarp +t−1 for updating the geometry. Rendering +Rg +t−1 used for updating the geometry operates on surfel- +level granularity, whereas Ra +t−1 used for geometry alignment +rendering operates on deformation node granularity. Another +difference between Rg +t−1 and Ra +t−1 is resolution, Rg +t−1 is up- +sampled by 4 × 4 compared to Ra +t−1 to prevent different +surfels from being projected onto the same pixel. Rg +t−1 +requires a higher resolution for accurate geometry update +and Ra +t−1 has a lower resolution for faster optimization. +2) Optical Flow: +The optical flow module receives +Ca +t−1,V a +t−1 from geometry rendering Ra +t−1, and Cm +t−1,V m +t−1 from +measurement Mt, and generates an optical-flow map OFt. +OFt predicts the optical-flow from previous geometry St−1 to +the latest measurement Mt. This prediction is later used for +registration through non-rigid alignment. We generate OFt +using a neural network based on the RAFT architecture [8], +along with additional global motion features as performed +in GMA [9]. The global motion features provide stability +for predicting motion features, even in occluded scenes. +Both RAFT and GMA models were originally trained using +only RGB images. The optical flow model used in [7] +shows that using RGB-D images for training provides a +far more stable flow, even with motion blurring. Thus, our +model is also trained on RGB-D images from the datasets +FlyingThings3D [26], Monkaa [26] and Sintel [27], [28]. +3) Non-rigid alignment: +Non-rigid alignment is per- +formed in order to compute non-rigid deformation Wt. This +step solves a massive optimization problem to warp the +previous geometry St−1 to a geometry Swarp +t−1 that fits current +measurement Mt. We use a Gauss-Seidel solver implemented +with CUDA to solve this problem, which is summarized as +minWEtotal(W) = wpicpEpicp(W)+w2DE2D(W) ++waregEareg(W), + +Mt +Mt +Lm +Input +Output +Sync +Sync +Measurement +St +(St Gt L) +-warp +Lm +-warp +R +Geometry +St-1 +St-1 +warp +Ls +Gt +Normal +Vertex +Vertex +Normal +va-1 +Ng +Sync +Sync +t-1 +Color +Index +Color +Index +Cf-1 +19-1where wpicp,wof ,wareg are the weights of terms Epicp(W), +E2D(W) and Eareg(W), explained in the following. +Registration. Let u = (x,y) be a pixel in measurement map +u, and let mi = M(u) bet its associated surfel. Let (mi,sΠ(i)) +denote a pair of registered measurement and geometry surfel. +Π(i) is defined as Π(i) := Ia +t−1(x − of x +t ,y − of y +t ), wherein +(x,y) = u,(o f x +t ,o f y +t ) = OFt(u). Ia +t−1 ∈ Ra +t−1 is the index map +of the rendered geometry. +PICP Loss. Point-to-point ICP loss is sensitive to dis- +turbance and outliers, which limits its utility in real-world +applications. Instead, we use a plane-based ICP (PICP) loss +to align the differences along the depth direction as follows, +Epicp(W) = ∑ +mi∈M +nm +i ·( ¯W(sΠ(i))vs +Π(i) −vm +i ), +(3) +wherein vs +Π(i) is the 3D coordinates of surfel sΠ(i), vm +i ,nm +i are +the 3D coordinates and normal of measurement surfel mi. ¯W +is defined in Eq. 2. +2D Loss. One limitation of the PICP loss is that it cannot +correctly capture motions within the same plane, such as +the moving calendar shown in Fig. 6. We thus add to the +objective function a 2D loss E2D defined as follows, +P = +� +1 +0 +0 +0 +1 +0 +� +(4) +E2D(W) = ∑ +mi∈M +∥P( ¯W(sΠ(i))vs +Π(i) −vm +i )∥2. +(5) +Here, P is a projection matrix, projecting the 3D difference +to the camera X-Y plane. This term constrains mi and sΠ(i) +to be as close as possible on the camera X-Y plane. It is +worth noting that our proposed 2D loss is different from the +one proposed in OcclusionFusion [7], where pixel differences +are used to calculate the 2D loss. The influence of pixel +differences scales with the distance to the camera, which +makes the optimization parameters harder to tune. +Semantic-aware Adaptive Deformation Graph. The tra- +ditional Embedded Deformation graph (ED-graph) has been +widely used in non-rigid tracking and non-rigid reconstruc- +tion. It can describe complicated warping fields with a simple +data structure and an interpolation strategy. Moreover, the as- +rigid-as-possible (ASAP) regulation term defined on defor- +mation nodes provides a continuity guarantee for neighboring +nodes. However, the ED-graph cannot handle the topology +changes of different nodes. For example, if we use an ED +graph to describe a cup being lifted up from a table, as shown +in Fig. 2 (a), the motion of the cup’s nodes also influences +and propagates to the table’s nodes. Many previous works +have attempted to overcome this limitation of the ED-graph +by proposing a dual deformation graph [18] or a level- +set-based TSDF fusion mechanism [11]. However, these +approaches are too complicated or introduce too much over- +head computation. We propose the Semantic-aware Adaptive +Deformation Graph (SAD-graph) to address this issue of +topological changes among objects with different semantic +classes. More importantly, the proposed algorithm is intuitive +and requires little extra computation compared with ED- +graph. Another advantage is that existing ED-graph based +approaches can be easily upgraded to support SAD-graph. +The core idea of SAD-graph is that instead of imposing a +uniform regulation continuity constraint on all deformation +nodes, constraints of varying adaptive strengths are imposed +on different edges. A variable weight ωi, j is associated with +the constraint (defined in Eq. 7) between neighboring nodes +gi and gj, and the strength of the constraint is systematically +adjusted. Constraint weight ωgi,gj is a function of lgi,lg j, the +semantic labels of nodes gi and gj. It is defined as follows, +ωgi,g j(lgi,lg j) = +� +0.1, +if lgi ̸= lg j +δ k, +if lgi = lg j = k,k ∈ [1,...,H +1] +(6) +where δ k is a constant describing the average rigidness of +objects belonging to semantic category k, e.g., δtable = 1.0, +δ human = 0.3, etc. For example, in Fig. 2 (b), since the +internal rigidity constraint within the cup or the table is much +larger than the constraint between them, their geometries can +be accurately reconstructed during the topology separation. +It is important to note that these constraints are not hard +because the semantic labels obtained from a neural network +detector are error-prone. +Fig. 2: Illustration of the proposed Semantic-aware Adaptive De- +formation Graph (SAD-graph). The scene describes a cup (brown) +being lifted up from the table (green). Black edges indicate strong +continuity constraints, while grey edges indicate weak constraints. +Adaptive Regulation Loss. We introduce a deformation +graph, SAD-graph, and a new regulation term, adaptive +regulation Eareg(W). A semantic-related connection weight +ω is used to adjust the regulation strength among and within +different semantic classes as follows, +Eareg(W) = ∑ +g j∈G +∑ +gi∈NG(g j) +ωgi,gj +��Tjpj −Tipi +��2 +2 , +(7) +wherein G is the deformation graph, NG(gj) refers to the set +of neighbors of node gj in the deformation graph, Tj and Ti +are the transformations defined on nodes gi and gj. pi and pj +are the 3D coordinate of gi and gj, and ωgi,g j is the weight +of the connection between nodes gi and gj, defined in Eq. 6. +4) Geometry and Graph Update: Once the non-rigid +deformation is computed, the geometry update process of +Mono-STAR is similar to SurfelWarp [10]. Thus, we only +briefly describe that process and we focus on the semantic +update. This step returns the updated geometry St (the +reconstructed scene) and the updated graph Gt, both of which +are needed for processing the scene in the next time-step. +Updating the Geometry. The previous geometry St−1 is +warped to Swarp +t−1 after the non-rigid alignment step. Although +Swarp +t−1 +is already close enough to measurement Mt, there + +Before deformation +After deformation +Before deformation +After deformation +区 +X +区 +(a) ED-graph +(b) SAD-graphstill exists a discrepancy between them due to measurement +noises, emerging surfaces, topology changes, or even track- +ing failures. The geometry is updated to address this gap +between the warped geometry Swarp +t−1 +and measurement Mt. +There are four steps in total in this process. +1. Registration: A projective registration is made between +measurement Mt and warped geometry Swarp +t−1 +according to +rendering map Rg +t−1. +2. Fusion: If a surfel mi ∈ Mt is mapped to s j ∈ Swarp +t−1 +in +the registration, mi is merged into sj to average measurement +noises. The semantic label ls j of sj is defined as a probability +distribution psj. When mi is fused into sj, ps j is also updated +by lmi. The update formula for psj is: +psj(k) = (psj(k)+δm)/∑ +k′ +psj(k′),if k = lmi +(8) +psj(k) = ps j(k)/∑ +k′ +psj(k′),otherwise. +(9) +Here, δm is the confidence of the measurement. +3. Append: If there are no surfels in Swarp +t−1 +that can be +registered to mi, mi must belong to a newly observed surface +or be noise. In the first case, mi will be appended to Swarp +t−1 . +The semantic label distribution ps +i of mi is initialized as: +ps +i(k) = δm,if k = lmi; ps +i(k) = 0,otherwise, +(10) +4. Removal: After each mi ∈ Mt is either fused or ap- +pended, some surfels sj ∈ Swarp +t−1 +are left with no correspon- +dence. A geometry violation test is performed on the remain- +ing surfels, and those that fail the test are removed [10]. +After the four steps given above, we get the updated +geometry St for time-step t. +Updating the Graph. The update of the SAD-graph is +identical to the update of the traditional ED-graph. The +update appends new nodes but does not remove existing ones. +Let Sappend be the set of the appended surfels during the +geometry update. We first compute the distances between +every surfel s ∈ Sappend and every node g ∈ Gt−1. Let +D(s,G) = ming∈G distance(s, g). A surfel s is said to be +unsupported if D(s,G) > σ, for some threshold σ. We per- +form a spatially uniform sampling from all the unsupported +surfels. Sampled surfels are appended to graph G as new +nodes. The semantic label of node gi, lgi is updated according +to the semantic labels of NS(gi), neighbor surfels of gi. +lgi = argmaxk{∑sj∈Ns(gi) δ(lsj,k)}. Here δ(ls j,k) = 1,if lsj = +k;δ(ls +j,k) = 0,otherwise. +V. EXPERIMENTS +We +test +our +technique +on +a +dataset +we +collected +and a public dataset VolumeDeform [29]. An ablation +study +and +comparisons +with +SoTA +methods +such +as +STAR-no-prior [2] and MaskFusion [4] on challenging +scenes are presented in this section. Since collecting +ground-truth geometry and deformation for non-rigid ob- +jects is extremely challenging, experiments and compar- +isons in this area are limited to qualitative results [29]. +Supplementary results and resources can be found at +https://github.com/changhaonan/Mono-STAR-demo. +A. Performance +We tested our system on a desktop machine with a +GeForce RTX 3090 and an AMD-Ryzen 9 5900X. On aver- +age, measurement loading takes 4 ms and segmentation costs +10 ms (Segmenter Mask [20]). The optimization module uses +20 ms. The geometry update uses 7 ms. The major bottleneck +is the optical-flow network, which takes 60 ms. Since the +latency for the measurement thread is fully hidden by the +geometry thread, our entire system runs in 11 Hz. If the +optical-flow runs on a separate graphic card, it would take +only 26 ms [7], which would double the speed of our system. +B. Qualitative Results +1) Soft objects: Fig. 3 illustrates the non-rigid deforma- +tion ability of MaskFusion and Mono-STAR. We can clearly +see that MaskFusion fails to track the deformations of the +pillow and umbrella, while Mono-STAR correctly captures +both of them in the reconstructed model, which shows the +advantage of our technique over MaskFusion in handling +non-rigid deformation. +Fig. 3: Comparison on deformable objects with MaskFusion on +our recorded dataset (top) and VolumeDeform dataset (bottom). +2) Fast Motion: Fig. 4 demonstrates Mono-STAR’s abil- +ity to handle fast motions. The top scene in Fig. 4 shows +an accident that was recorded during our data collection. +While we were pushing a cup on the table, the cup hit a +bump and fell down. The bottom scene is about passing a +basketball between two hands. Objects in both scenes moved +very fast. One is 18 frames, and the other is 30 frames. +Significant motion blur can be observed in both middle +images. However, Mono-STAR can still capture these fast +motions and correctly reconstruct the objects at each frame. + += += +150 +Measurement +MaskFusion +OursFig. 4: Experiment on fast motion. Pushing down a coffee cup +(top). Passing a basketball between two hands (bottom). The second +and fourth row are our 3D scene reconstruction results. +3) Resilience to Semantic Segmentation Noises: Fig. 5 +shows how our proposed method can resist noise in semantic +segmentation. The figures on the left are the RGB measure- +ment from the beginning and the end frames. The right side +compares the segmentation from the measurement and the +segmentation from our reconstruction. Although the ground- +truth measurement suffers from major segmentation errors, +where the cup label is completely lost for t > 0, Mono-STAR +still maintains the correct semantic labels in its reconstruction +result through semantic fusion. +Fig. 5: Resilience to semantic label noise. The top sequence is the +segmentation map Lmt of the measurement. The bottom sequence is +the segmentation map Lst from our reconstruction technique. +C. Ablation Study +1) 2D Loss: We test Mono-STAR with and without the 2D +loss on the “adventcalender” dataset from VolumeDeform. +Fig. 6 shows that the proposed 2D loss E2D can efficiently +track the motions within a plane. In contrast, tracking without +E2D fails in this type of motion, which clearly shows the +effectiveness of the proposed 2D loss. +2) SAD-graph: In Fig. 7, we compare the ED-graph with +the topology-aware ED-graph (STAR-no-prior) and the SAD- +graph (Mono-STAR). We can see that the ED-graph fails +Fig. 6: Ablation study on the 2D loss. +to support the topology change that results from lifting the +object from the table. Topology-aware ED-graph can separate +the topology, but it also generates many outliers on the table. +With the help of the proposed SAD-graph, Mono-STAR can +conduct a smoother and cleaner separation. +Fig. 7: Comparing ED-graph (left), topology-aware ED-graph +(middle, STAR-no-prior [2]), and SAD-graph (right, ours). The +scene shows a plushy toy being lifted up from the table. +D. Discussion of Limitations +Although Mono-STAR shows great potential in many dif- +ferent aspects, it still has two limitations. First, it relies on the +optical flow to track fast motions. However, even the state-of- +art optical flow detector GMA [9] is not always accurate, es- +pecially when the motion is too fast and the tracked surfaces +are heavily occluded. Our system can tolerate some noise +from the GMA optical-flow module. However, if the optical +flow provides inaccurate predictions for multiple consecutive +frames, the tracking of the corresponding object may still +fail. Another drawback of our system is the incompleteness +of the reconstructed geometry. Our reconstructed geometries +usually have holes and are not as smooth as TSDF-based +geometry. The reason is that Surfel-based geometry, unlike +TSDF-based geometry, is discrete by default. Therefore, +it is difficult to maintain the smoothness of Surfel-based +geometry in highly dynamic scenes. These two challenges +can be addressed in future works. +VI. CONCLUSION +We presented Mono-STAR, a single-view solution for +the semantic-aware STAR problem. Mono-STAR uses a +novel semantic-aware and adaptive deformation graph for +simultaneous tracking and reconstruction, and can handle +topology changes as well as semantic fusion. Experiments +show that Mono-STAR achieves promising results in non- +rigid object reconstruction, while resisting to semantic seg- +mentation errors, and capturing fast motions on various chal- +lenging scenes. We believe that this system can inspire and +boost more future research on imitation learning, dexterous +manipulation, and many other relevant robotics problems. + +t = 47 +t = 58 +t = 65 +t = 15 +t = 26 +t = 45Human +Cup +Background +Measurement +t=0 +t=0 +t = 102 +t = 162 +Reconstruction +t = 162 +0=1 +t = 102 +t = 162t= 1 +t = 41 +t = 41 +Initial +without 2D Loss +Ourst =31 +t =31 +t = 38 +t = 38 +ED-graph +STAR-no-prior +OursREFERENCES +[1] B. Wen, W. Lian, K. Bekris, and S. 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Brox, “A large dataset to train convolutional networks +for disparity, optical flow, and scene flow estimation,” in IEEE +International Conference on Computer Vision and Pattern Recognition +(CVPR), 2016, arXiv:1512.02134. [Online]. Available: http://lmb. +informatik.uni-freiburg.de/Publications/2016/MIFDB16 +[27] D. J. Butler, J. Wulff, G. B. Stanley, and M. J. Black, “A naturalistic +open source movie for optical flow evaluation,” in European Conf. on +Computer Vision (ECCV), ser. Part IV, LNCS 7577, A. Fitzgibbon et +al. (Eds.), Ed. +Springer-Verlag, Oct. 2012, pp. 611–625. +[28] J. Wulff, D. J. Butler, G. B. Stanley, and M. J. Black, “Lessons +and insights from creating a synthetic optical flow benchmark,” in +ECCV Workshop on Unsolved Problems in Optical Flow and Stereo +Estimation, ser. Part II, LNCS 7584, A. Fusiello et al. (Eds.), Ed. +Springer-Verlag, Oct. 2012, pp. 168–177. +[29] M. +Innmann, +M. +Zollh¨ofer, +M. +Nießner, +C. +Theobalt, +and +M. +Stamminger, +“VolumeDeform: +Real-time +Volumetric +Non- +rigid +Reconstruction,” +mar +2016. +[Online]. +Available: +http: +//arxiv.org/abs/1603.08161 + diff --git a/29FQT4oBgHgl3EQfGTVE/content/tmp_files/load_file.txt b/29FQT4oBgHgl3EQfGTVE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c820a28b05d971e2b5c29bcbeadbf148fd192364 --- /dev/null +++ b/29FQT4oBgHgl3EQfGTVE/content/tmp_files/load_file.txt @@ -0,0 +1,664 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf,len=663 +page_content='Mono-STAR: Mono-camera Scene-level Tracking and Reconstruction Haonan Chang1, Dhruv Metha Ramesh1, Shijie Geng1, Yuqiu Gan, Abdeslam Boularias1 Abstract— We present Mono-STAR, the first real-time 3D reconstruction system that simultaneously supports semantic fusion, fast motion tracking, non-rigid object deformation, and topological change under a unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The proposed system solves a new optimization problem incorporating optical-flow-based 2D constraints to deal with fast motion and a novel semantic-aware deformation graph (SAD-graph) for handling topology change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' We test the proposed system under various challenging scenes and demonstrate that it significantly outperforms existing state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Supplementary material, including videos, can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='com/changhaonan/Mono-STAR-demo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' INTRODUCTION Real-time perception is a crucial component of modern robotic manipulation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Recently, You Demonstrate Only Once [1] has demonstrated that given the geometry model and 6D-pose trajectory of a manipulated object during an expert demonstration, a robot can quickly learn complex and contact-rich manipulation skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Such progress shows the importance of geometric 3D reconstruction and tracking systems for robotic manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' However, a perception system that can perform both track- ing and reconstruction simultaneously is notoriously difficult to build because reconstruction and tracking inherently de- pend on each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' For example, tracking algorithms usually require geometry models, while dynamic scene reconstruc- tion relies on accurate tracking for producing those geometry models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Scene-level Tracking and Reconstruction (STAR) [2] refers to a category of perception systems that generate both the geometry and the pose of every visible object in a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' This problem is related to the multiple-instance dynamic SLAM problem, where all movable objects in the scene are assumed to be rigid so that the problem can be decomposed into multiple dense-SLAM sub-problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' This approach was proposed in Co-Fusion [3] and MaskFusion [4], where a semantic neural network was employed first to decompose the scene into multiple objects and then deal with each object individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' This approach requires every object in the scene to be rigid or quasi-rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The same problem was investigated in MidFusion [5], where an octree was used to improve reconstruction and tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' However, these systems are limited to scenes of rigid objects with slow motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Instead of dealing with each object individually based on their semantic labels, one can also reconstruct all the objects in the scene as one large non-rigid object and segment them ulteriorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' This approach was however very challenging 1 Authors are with the Department of Computer Science, Rutgers University, 08854 New Brunswick, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' This work is supported by NSF awards 1734492, 1846043, and 2132972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' to apply until the introduction of the first real-time non- rigid reconstruction DynamicFusion [6], where the non- rigid reconstruction problem was decomposed into two sub- problems, (1) building the geometry at the initial frame, and (2) computing the deformation using an embedded defor- mation graph, namely ED-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' This paradigm was also followed in OcclusionFusion [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Inspired by these previous efforts, a solution to the general STAR problem was recently proposed in STAR-no-prior [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' In contrast to SLAM-based methods, STAR-no-prior reverses the order of segmentation and reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The entire scene is first reconstructed and then segmented into different objects based on topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' By doing so, STAR-no-prior outperforms previous state-of-the- art methods such as [4] and MidFusion [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' However, a major limitation of STAR-no-prior is its re- liance on a system of multiple cameras surrounding the scene, making it impractical for a mobile robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' To address this shortcoming, we propose Mono-STAR, a mono-camera STAR solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Switching from a multi-camera system to a mono-camera setting requires solving several non-trivial problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Notably, STAR-no-prior relies on the multi-camera system to overcome the plane-based-ICP constraint that it inherits from DynamicFusion, which supports tracking of only slow motion along the camera view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The use of multiple cameras can guarantee that any motion has at least one non- zero projection to a camera view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' However, the mono-camera setting does not have such a guarantee and therefore requires a new solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Occlusion Fusion [7] adds a 2D constraint using optical-flow (RAFT [8] or GMA [9]) to deal with fast motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Inspired by this, we propose a new 2D loss to track motions that are perpendicular to the camera view, which not only stabilizes tracking performance under a single view but also improves our system’s ability to handle fast motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Furthermore, STAR-no-prior does not take advantage of semantic labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' We, therefore, combine the semantic infor- mation with the embedded deformation graph mechanism and propose a Semantic-aware Adaptive Deformation graph, SAD-graph, which is an extension of ED-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' With just lit- tle extra computation, SAD-graph can easily handle topology changes across distinct semantic classes and assign different levels of rigidness for each type of object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' To the best of our knowledge, Mono-STAR is the first single-view real-time 3D reconstruction system that can simultaneously handle seman- tic fusion, fast motion tracking, non-rigid object deformation, and topological change under one unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' RELATED WORKS Simultaneous Tracking and Reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Simulta- neous 6D tracking and 3D reconstruction was typically arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='13244v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='RO] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='30 Jan 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='Method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='Semantic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='Fast ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='Non-rigid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='Topology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='Single ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='motion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='objects ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='change ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='view ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='SLAM++ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='DynamicFusion [6] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='Volume Deform ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='SurfelWarp [10] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='TCAFusion [11] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='Co-fusion [3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='Fusion4D [12] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='Motion2Fusion [13] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='Functon4D [14] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='MaskFusion [4] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='RigidFusion [15] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='MidFusion [5] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='OcclusionFusion [7] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='STAR-no-prior [2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='Mono-STAR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='\x14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='TABLE I: Taxonomy of the-state-of-art scene-level fusion systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' regarded in previous works as a multiple-instance dynamic SLAM problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Many works such as Co-fusion [3], Mask- Fusion [4], and RigidFusion [15] proposed to divide the scene into multiple rigid objects and track each object individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' More recently, STAR-no-prior [2] formalized the STAR problem as a scene-level non-rigid reconstruction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Our mono-camera system eliminates the multi- camera requirement of STAR-no-prior by adding a new optical-flow-based 2D constraint and a novel semantic-aware adaptive deformation graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Dynamic Scene Reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Dynamic scene recon- struction [16], [17] is the problem of reconstructing the geometry and recording the deformation of a scene with moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' DynamicFusion [6] was the first real-time GPU-based solution for solving this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' It adopts a TSDF-based geometry as the canonical model and an embed- deformation graph (ED-graph) to describe the deformation of the whole scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' A drawback of this method is that the combination of TSDF and ED-graph cannot handle topology changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Many recent techniques such as Fusion4D [12], Motion2Fusion [13], [11], [18] have attempted to address this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' However, these methods require significantly more computation or rely on expensive sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' SurfelWarp [10] demonstrated that a Surfel-based representation can be used to tackle topology changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Therefore, our proposed system also adopts a Surfel-based representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' PROBLEM FORMULATION AND BACKGROUND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Problem formulation Given a sequence of RGB-D images of a given dynamic scene taken from a single fixed camera, we consider the problem of simultaneous tracking and reconstruction of all the objects visible in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The number of objects is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The objects can be non-rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Measurement, Mt can defined as set of measurement surfels mi at time-step t, generated from the RGB-D input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' mi = (vi,ni,ci), where v,n,c are 3D coordinates, normal and color respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The proposed system returns at each time-step t a Surfel- based geometry St (the reconstructed scene) for the entire scene and its corresponding deformation graph Gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Surfel- based geometry St is a set of surfels si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' si = (vi,ni,ci,ri,lsi), where vi,ni,ci,ri,lsi are respectively the 3D coordinates, normal, color, radius and semantic label of surfel si ∈ St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' We assume that there is a maximum of H pre-defined different semantic categories {1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=',H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' If a surfel does not belong to any pre-defined category, it will be labeled as H +1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=', unrecognized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Deformation graph Gt is defined by a set of nodes {gi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Each node gi has a semantic label lgi, and is connected to its nearest-neighbor nodes, denoted as NG(gi), in the 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Deformation graph Gt is associated with a warp field Wt, defined as W = {[pi ∈ R3,δi ∈ R+,Ti ∈ SE(3)]}, wherein i is the index of a node in Gt, pi is the 3D point that corresponds to node gi, δi is the node’s radius of influence, and Ti is the 6D transformation defined on node gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Ti is represented by a dual quaternion qi for smooth interpolation [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Warp field W describes the deformation between two consecutive time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' For each surfel s = (v,n,c,r,l) ∈ S, we compute its 6D transformation ¯W(s) based on warp field W, ¯W(s) = normalize( ∑ k∈NG(s) w(v, pk)qk), (1) wherein NG(s) denotes the neighbors nodes of surfel s, w(s) is an interpolation parameter, defined as w(s) = exp � ∥v− pk∥2 2 /(2δ 2 k ) � , and v is the 3D position of surfel s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The local transformation ¯W(s) is then used to describe the deformation of surfel s as follows: vwarp = ¯W(s)v nwarp = rotation � ¯W(s) � n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' (2) Here, v,n are the vertex and normal of s before warping, and vwarp,nwarp are the vertex and normal after the deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Symbol Meaning Definition Mt Measurement at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' III-A, IV-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='1 St−1 Surfel geometry from t-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' III-A Ra t−1 2D maps rendered from St−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='1 Swarp t−1 Warped geometry after non-rigid alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='4 Rg t−1 2D maps rendered from Swarp t−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='1 TABLE II: Notation sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' PROPOSED APPROACH An overview of the proposed method is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Mono-STAR uses two parallel threads, a measurement thread, and a geometry thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The first thread is responsible for loading measurements Mt and generating semantic labels Lm t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The geometry thread uses this measurement Mt and the alignment rendering Ra t−1 to compute an optical-flow OFt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Then, Mt,Ra t−1,OFt are given to the optimization module that then computes the non-rigid deformation Wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' After the non- rigid alignment, previous geometry St−1 is warped to Swarp t−1 , and the geometry rendering Rg t−1 is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Finally, Rg t−1, deformation Wt, warped geometry Swarp t−1 , and semantic labels Lm t are combined to generate the latest geometry St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Noticeably, loading Mt takes less time than updating geometry St−1 and rendering Ra t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Semantic segmentation is also faster than the combined process of generating optical- flow OFt and the non-rigid alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Thus, the geometry thread fully hides the latency of the measurement thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 1: Overview of the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The system runs in two parallel threads, one for measurement and one for geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' In each time-step t, the measurement thread loads a measurement Mt from images or a camera buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Then, a segmentation network generates a set of semantic labels Lmt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Once the measurement is loaded on the GPU memory, Mt and previous alignment rendering Ra t−1 are fed into an optical-flow network to generate the optical-flow OFt from previous geometry St−1 to measurement Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Optical-flow OFt, geometry rendering Rt and measurement Mt are used to compute warp-field Wt with non-rigid alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' After the alignment, previous geometry St−1 will be warped to Swarp t−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The fusion rendering map Rg t−1 is then rendered from Swarp t−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Rg t−1, Swarp t−1 and semantic labels Lmt are used to generate the updated geometry St, deformation graph Gt and the surfel semantic label Lst .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Measurement Thread 1) Measurement: We use one Intel RealSense-415 camera to collect RGB-D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Depth images are denoised with a Gaussian filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The maximum frame rate for this module is limited to 20 fps to coordinate with other modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' We use a double-buffer strategy to hide latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Specifically, we use two buffers B0 and B1 to store measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' When B0 is used by other threads, B1 can read images simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The filtered images are used to construct three maps, V m t , Nm t , Cm t , storing 3D coordinates vm i , normal nm i , and color cm i , respectively, for surfel mi of each pixel in the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 2) Segmentation: The segmentation module receives the color map Cm t and returns a semantic label map Lm t of H pre-defined semantic classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Here, we use two different segmentation models, a transformer-based Segmenter Mask [20], and a more traditional MaskRCNN [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The two models are pre-trained on two different datasets, ADE20K [22], [23] and COCO-Stuff [24] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' We do not further train these models on any other dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' We select which one to use based on the types of objects in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Geometry Thread 1) Geometry Rendering: The input of the geometry ren- dering pipeline is a geometry S, and the output is the rasterized rendering R for geometry S from the current camera view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The rendering process to generate R follows the classical point cloud rasterizing process [25], where every surfel is projected to its nearest pixel position on the camera plane based on its 3D coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Each rendering map R is composed of four 2D maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Rt = {Ct,Vt,Nt,It}, where Vt,Nt,Ct,It are respectively the vertex map, the normal map, the color map and the index map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' These maps store the 3D coordinates vi, the normal ni, the color ci and the surfel index i of the projected surfel si at each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' At each time-step, the geometry rendering pipeline is called twice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' once to generate Ra t−1 from previous geometry St−1 for non-rigid alignment, and once to get Rg t−1 with warped geometry Swarp t−1 for updating the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Rendering Rg t−1 used for updating the geometry operates on surfel- level granularity, whereas Ra t−1 used for geometry alignment rendering operates on deformation node granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Another difference between Rg t−1 and Ra t−1 is resolution, Rg t−1 is up- sampled by 4 × 4 compared to Ra t−1 to prevent different surfels from being projected onto the same pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Rg t−1 requires a higher resolution for accurate geometry update and Ra t−1 has a lower resolution for faster optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 2) Optical Flow: The optical flow module receives Ca t−1,V a t−1 from geometry rendering Ra t−1, and Cm t−1,V m t−1 from measurement Mt, and generates an optical-flow map OFt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' OFt predicts the optical-flow from previous geometry St−1 to the latest measurement Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' This prediction is later used for registration through non-rigid alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' We generate OFt using a neural network based on the RAFT architecture [8], along with additional global motion features as performed in GMA [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The global motion features provide stability for predicting motion features, even in occluded scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Both RAFT and GMA models were originally trained using only RGB images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The optical flow model used in [7] shows that using RGB-D images for training provides a far more stable flow, even with motion blurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Thus, our model is also trained on RGB-D images from the datasets FlyingThings3D [26], Monkaa [26] and Sintel [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 3) Non-rigid alignment: Non-rigid alignment is per- formed in order to compute non-rigid deformation Wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' This step solves a massive optimization problem to warp the previous geometry St−1 to a geometry Swarp t−1 that fits current measurement Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' We use a Gauss-Seidel solver implemented with CUDA to solve this problem, which is summarized as minWEtotal(W) = wpicpEpicp(W)+w2DE2D(W) +waregEareg(W), Mt Mt Lm Input Output Sync Sync Measurement St (St Gt L) warp Lm warp R Geometry St-1 St-1 warp Ls Gt Normal Vertex Vertex Normal va-1 Ng Sync Sync t-1 Color Index Color Index Cf-1 19-1where wpicp,wof ,wareg are the weights of terms Epicp(W), E2D(W) and Eareg(W), explained in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Let u = (x,y) be a pixel in measurement map u, and let mi = M(u) bet its associated surfel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Let (mi,sΠ(i)) denote a pair of registered measurement and geometry surfel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Π(i) is defined as Π(i) := Ia t−1(x − of x t ,y − of y t ), wherein (x,y) = u,(o f x t ,o f y t ) = OFt(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Ia t−1 ∈ Ra t−1 is the index map of the rendered geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' PICP Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Point-to-point ICP loss is sensitive to dis- turbance and outliers, which limits its utility in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Instead, we use a plane-based ICP (PICP) loss to align the differences along the depth direction as follows, Epicp(W) = ∑ mi∈M nm i ·( ¯W(sΠ(i))vs Π(i) −vm i ), (3) wherein vs Π(i) is the 3D coordinates of surfel sΠ(i), vm i ,nm i are the 3D coordinates and normal of measurement surfel mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' ¯W is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 2D Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' One limitation of the PICP loss is that it cannot correctly capture motions within the same plane, such as the moving calendar shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' We thus add to the objective function a 2D loss E2D defined as follows, P = � 1 0 0 0 1 0 � (4) E2D(W) = ∑ mi∈M ∥P( ¯W(sΠ(i))vs Π(i) −vm i )∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' (5) Here, P is a projection matrix, projecting the 3D difference to the camera X-Y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' This term constrains mi and sΠ(i) to be as close as possible on the camera X-Y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' It is worth noting that our proposed 2D loss is different from the one proposed in OcclusionFusion [7], where pixel differences are used to calculate the 2D loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The influence of pixel differences scales with the distance to the camera, which makes the optimization parameters harder to tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Semantic-aware Adaptive Deformation Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The tra- ditional Embedded Deformation graph (ED-graph) has been widely used in non-rigid tracking and non-rigid reconstruc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' It can describe complicated warping fields with a simple data structure and an interpolation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Moreover, the as- rigid-as-possible (ASAP) regulation term defined on defor- mation nodes provides a continuity guarantee for neighboring nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' However, the ED-graph cannot handle the topology changes of different nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' For example, if we use an ED graph to describe a cup being lifted up from a table, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 2 (a), the motion of the cup’s nodes also influences and propagates to the table’s nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Many previous works have attempted to overcome this limitation of the ED-graph by proposing a dual deformation graph [18] or a level- set-based TSDF fusion mechanism [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' However, these approaches are too complicated or introduce too much over- head computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' We propose the Semantic-aware Adaptive Deformation Graph (SAD-graph) to address this issue of topological changes among objects with different semantic classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' More importantly, the proposed algorithm is intuitive and requires little extra computation compared with ED- graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Another advantage is that existing ED-graph based approaches can be easily upgraded to support SAD-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The core idea of SAD-graph is that instead of imposing a uniform regulation continuity constraint on all deformation nodes, constraints of varying adaptive strengths are imposed on different edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' A variable weight ωi, j is associated with the constraint (defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 7) between neighboring nodes gi and gj, and the strength of the constraint is systematically adjusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Constraint weight ωgi,gj is a function of lgi,lg j, the semantic labels of nodes gi and gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' It is defined as follows, ωgi,g j(lgi,lg j) = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='1, if lgi ̸= lg j δ k, if lgi = lg j = k,k ∈ [1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=',H +1] (6) where δ k is a constant describing the average rigidness of objects belonging to semantic category k, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=', δtable = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='0, δ human = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='3, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 2 (b), since the internal rigidity constraint within the cup or the table is much larger than the constraint between them, their geometries can be accurately reconstructed during the topology separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' It is important to note that these constraints are not hard because the semantic labels obtained from a neural network detector are error-prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 2: Illustration of the proposed Semantic-aware Adaptive De- formation Graph (SAD-graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The scene describes a cup (brown) being lifted up from the table (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Black edges indicate strong continuity constraints, while grey edges indicate weak constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Adaptive Regulation Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' We introduce a deformation graph, SAD-graph, and a new regulation term, adaptive regulation Eareg(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' A semantic-related connection weight ω is used to adjust the regulation strength among and within different semantic classes as follows, Eareg(W) = ∑ g j∈G ∑ gi∈NG(g j) ωgi,gj ��Tjpj −Tipi ��2 2 , (7) wherein G is the deformation graph, NG(gj) refers to the set of neighbors of node gj in the deformation graph, Tj and Ti are the transformations defined on nodes gi and gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' pi and pj are the 3D coordinate of gi and gj, and ωgi,g j is the weight of the connection between nodes gi and gj, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 4) Geometry and Graph Update: Once the non-rigid deformation is computed, the geometry update process of Mono-STAR is similar to SurfelWarp [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Thus, we only briefly describe that process and we focus on the semantic update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' This step returns the updated geometry St (the reconstructed scene) and the updated graph Gt, both of which are needed for processing the scene in the next time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Updating the Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The previous geometry St−1 is warped to Swarp t−1 after the non-rigid alignment step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Although Swarp t−1 is already close enough to measurement Mt, there Before deformation After deformation Before deformation After deformation 区 X 区 (a) ED-graph (b) SAD-graphstill exists a discrepancy between them due to measurement noises, emerging surfaces, topology changes, or even track- ing failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The geometry is updated to address this gap between the warped geometry Swarp t−1 and measurement Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' There are four steps in total in this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Registration: A projective registration is made between measurement Mt and warped geometry Swarp t−1 according to rendering map Rg t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Fusion: If a surfel mi ∈ Mt is mapped to s j ∈ Swarp t−1 in the registration, mi is merged into sj to average measurement noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The semantic label ls j of sj is defined as a probability distribution psj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' When mi is fused into sj, ps j is also updated by lmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The update formula for psj is: psj(k) = (psj(k)+δm)/∑ k′ psj(k′),if k = lmi (8) psj(k) = ps j(k)/∑ k′ psj(k′),otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' (9) Here, δm is the confidence of the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Append: If there are no surfels in Swarp t−1 that can be registered to mi, mi must belong to a newly observed surface or be noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' In the first case, mi will be appended to Swarp t−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The semantic label distribution ps i of mi is initialized as: ps i(k) = δm,if k = lmi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' ps i(k) = 0,otherwise, (10) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Removal: After each mi ∈ Mt is either fused or ap- pended, some surfels sj ∈ Swarp t−1 are left with no correspon- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' A geometry violation test is performed on the remain- ing surfels, and those that fail the test are removed [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' After the four steps given above, we get the updated geometry St for time-step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Updating the Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The update of the SAD-graph is identical to the update of the traditional ED-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The update appends new nodes but does not remove existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Let Sappend be the set of the appended surfels during the geometry update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' We first compute the distances between every surfel s ∈ Sappend and every node g ∈ Gt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Let D(s,G) = ming∈G distance(s, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' A surfel s is said to be unsupported if D(s,G) > σ, for some threshold σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' We per- form a spatially uniform sampling from all the unsupported surfels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Sampled surfels are appended to graph G as new nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The semantic label of node gi, lgi is updated according to the semantic labels of NS(gi), neighbor surfels of gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' lgi = argmaxk{∑sj∈Ns(gi) δ(lsj,k)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Here δ(ls j,k) = 1,if lsj = k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='δ(ls j,k) = 0,otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' EXPERIMENTS We test our technique on a dataset we collected and a public dataset VolumeDeform [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' An ablation study and comparisons with SoTA methods such as STAR-no-prior [2] and MaskFusion [4] on challenging scenes are presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Since collecting ground-truth geometry and deformation for non-rigid ob- jects is extremely challenging, experiments and compar- isons in this area are limited to qualitative results [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Supplementary results and resources can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content='com/changhaonan/Mono-STAR-demo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Performance We tested our system on a desktop machine with a GeForce RTX 3090 and an AMD-Ryzen 9 5900X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' On aver- age, measurement loading takes 4 ms and segmentation costs 10 ms (Segmenter Mask [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The optimization module uses 20 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The geometry update uses 7 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The major bottleneck is the optical-flow network, which takes 60 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Since the latency for the measurement thread is fully hidden by the geometry thread, our entire system runs in 11 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' If the optical-flow runs on a separate graphic card, it would take only 26 ms [7], which would double the speed of our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Qualitative Results 1) Soft objects: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 3 illustrates the non-rigid deforma- tion ability of MaskFusion and Mono-STAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' We can clearly see that MaskFusion fails to track the deformations of the pillow and umbrella, while Mono-STAR correctly captures both of them in the reconstructed model, which shows the advantage of our technique over MaskFusion in handling non-rigid deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 3: Comparison on deformable objects with MaskFusion on our recorded dataset (top) and VolumeDeform dataset (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 2) Fast Motion: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 4 demonstrates Mono-STAR’s abil- ity to handle fast motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The top scene in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 4 shows an accident that was recorded during our data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' While we were pushing a cup on the table, the cup hit a bump and fell down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The bottom scene is about passing a basketball between two hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Objects in both scenes moved very fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' One is 18 frames, and the other is 30 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Significant motion blur can be observed in both middle images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' However, Mono-STAR can still capture these fast motions and correctly reconstruct the objects at each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' = = 150 Measurement MaskFusion OursFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 4: Experiment on fast motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Pushing down a coffee cup (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Passing a basketball between two hands (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The second and fourth row are our 3D scene reconstruction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 3) Resilience to Semantic Segmentation Noises: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 5 shows how our proposed method can resist noise in semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The figures on the left are the RGB measure- ment from the beginning and the end frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The right side compares the segmentation from the measurement and the segmentation from our reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Although the ground- truth measurement suffers from major segmentation errors, where the cup label is completely lost for t > 0, Mono-STAR still maintains the correct semantic labels in its reconstruction result through semantic fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 5: Resilience to semantic label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The top sequence is the segmentation map Lmt of the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The bottom sequence is the segmentation map Lst from our reconstruction technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Ablation Study 1) 2D Loss: We test Mono-STAR with and without the 2D loss on the “adventcalender” dataset from VolumeDeform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 6 shows that the proposed 2D loss E2D can efficiently track the motions within a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' In contrast, tracking without E2D fails in this type of motion, which clearly shows the effectiveness of the proposed 2D loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 2) SAD-graph: In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 7, we compare the ED-graph with the topology-aware ED-graph (STAR-no-prior) and the SAD- graph (Mono-STAR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' We can see that the ED-graph fails Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 6: Ablation study on the 2D loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' to support the topology change that results from lifting the object from the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Topology-aware ED-graph can separate the topology, but it also generates many outliers on the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' With the help of the proposed SAD-graph, Mono-STAR can conduct a smoother and cleaner separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' 7: Comparing ED-graph (left), topology-aware ED-graph (middle, STAR-no-prior [2]), and SAD-graph (right, ours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The scene shows a plushy toy being lifted up from the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Discussion of Limitations Although Mono-STAR shows great potential in many dif- ferent aspects, it still has two limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' First, it relies on the optical flow to track fast motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' However, even the state-of- art optical flow detector GMA [9] is not always accurate, es- pecially when the motion is too fast and the tracked surfaces are heavily occluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Our system can tolerate some noise from the GMA optical-flow module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' However, if the optical flow provides inaccurate predictions for multiple consecutive frames, the tracking of the corresponding object may still fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Another drawback of our system is the incompleteness of the reconstructed geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Our reconstructed geometries usually have holes and are not as smooth as TSDF-based geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' The reason is that Surfel-based geometry, unlike TSDF-based geometry, is discrete by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Therefore, it is difficult to maintain the smoothness of Surfel-based geometry in highly dynamic scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' These two challenges can be addressed in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' CONCLUSION We presented Mono-STAR, a single-view solution for the semantic-aware STAR problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Mono-STAR uses a novel semantic-aware and adaptive deformation graph for simultaneous tracking and reconstruction, and can handle topology changes as well as semantic fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' Experiments show that Mono-STAR achieves promising results in non- rigid object reconstruction, while resisting to semantic seg- mentation errors, and capturing fast motions on various chal- lenging scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' We believe that this system can inspire and boost more future research on imitation learning, dexterous manipulation, and many other relevant robotics problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} +page_content=' t = 47 t = 58 t = 65 t = 15 t = 26 t = 45Human Cup Background Measurement t=0 t=0 t = 102 t = 162 Reconstruction t = 162 0=1 t = 102 t = 162t= 1 t = 41 t = 41 Initial without 2D Loss Ourst =31 t =31 t = 38 t = 38 ED-graph STAR-no-prior OursREFERENCES [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf'} 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b/49AyT4oBgHgl3EQfpPiQ/content/tmp_files/2301.00522v1.pdf.txt @@ -0,0 +1,1359 @@ +arXiv:2301.00522v1 [math.RT] 2 Jan 2023 +Irreducible module decompositions of rank 2 +symmetric hyperbolic Kac-Moody Lie algebras by +sl2 subalgebras which are generalizations of +principal sl2 subalgebras +TSURUSAKI Hisanori∗ +Abstract +There exist principal sl2 subalgebras for hyperbolic Kac-Moody Lie +algebras. +In the case of rank 2 symmetric hyperbolic Kac-Moody Lie +algebras, certain sl2 subalgebras are constructed. These subalgebras are +generalizations of principal sl2 subalgebras. +We show that the rank 2 +symmetric hyperbolic Kac-Moody Lie algebras themselves are irreducibly +decomposed under the action of this sl2 subalgebras. Furthermore, we +classify irreducible components of the decomposition. In particular, we +obtain multiplicities of unitary principal series and complementary series. +1 +Introduction +A nilpotent orbit in a finite dimensional simple Lie algebra g0 is an orbit ob- +tained by acting on the nilpotent element x of g0 by inner automorphisms. In +[Dyn57], these are classified by weighted Dynkin diagrams. From the Jacobson- +Morozov theorem, for a nilpotent element x of g0, we can construct a sl2-triple +with x as a nilpositive element ([CM93, Theorem 3.3.1]). This makes it equiva- +lent to classify nilpotent orbits of g0 and to classify sl2 triples in g0 up to inner +automorphisms. Among the nilpotent orbits of a finite dimensional simple Lie +algebra, the one whose dimension as an algebraic variety is maximal is called the +principal nilpotent orbit. Correspondingly, we can construct a principal SO(3) +subalgebra that is compatible with compact involution ([Kos59]). +Kac-Moody Lie algebras are generalizations of finite-dimensional simple Lie +algebras. +They are classified into three types: finite type, affine type, and +indefinite type. The finite type Kac-Moody Lie algebras are finite dimensional +simple Lie algebras. Within indefinite Kac-Moody Lie algebra, there is a class +called hyperbolic Kac-Moody Lie algebra. A hyperbolic Kac-Moody Lie algebra +∗Graduate +School +of +Mathematical +Sciences, +University +of +Tokyo +, +htsu- +rusaki1929@gmail.com +1 + +is an indefinite type Kac-Moody Lie algebra such that any true subdiagram of +its Dynkin diagram is of finite or affine type. +By analogy with the above theory, in [NO01], for a hyperbolic Kac-Moody +Lie algebra, its principal SO(1, 2) subalgebra was constructed. Note that [GOW02] +shows that it is possible to construct a principal SO(1, 2) subalgebra for certain +indefinite Kac-Moody Lie algebra that is not hyperbolic. +Corresponding to this principal SO(1, 2) subalgebra, we can construct a +principal sl2-subalgebra in a hyperbolic Kac-Moody Lie algebra. In [Tsu], for +the rank 2 symmetric hyperbolic Kac-Moody Lie algebras g, the following re- +sult is obtained. Let the space that the positive real root vectors span be Rg. +we consider sl2 subalgebras whose nilpositive element exists in Rg. Then we +can construct certain sl2 subalgebras. These subalgebras are generalizations of +principal sl2 subalgebra. +In this paper, for an sl2 subalgebra of rank 2 symmetric hyperbolic Kac- +Moody Lie algebra g constructed in [Tsu], we show g is decomposed into irre- +ducible sl2-modules by its action on g. +We are going to more details. Let s be an sl2 subalgebra constructed in +[Tsu]. Let H, X, Y be an sl2 triple and assume that s is spanned by H, X, Y . +Let the Chevalley generators of g be ei, fi, hi, (i = 0, . . . , n − 1). Let hR be the +R-span of Chevalley generators. From [Kac90, Theorem 2.2], g has a C-valued +nondegenerate invariant symmetric bilinear form (· | ·) called the standard form. +An antilinear automorphism ω0 of g, called compact involution, is determined +by +ω0(ei) = −fi, +ω0(fi) = −ei +(i = 0, . . . , n − 1), +ω0(h) = −h +(h ∈ hR). +From [Kac90, §2.7], we can determine a nondegenerate Hermitian form (· | ·)0 +on g with (x | y)0 = −(ω0(x) | y). +s-module V is called unitarizable if the following conditions are satisfied. +(1) (· | ·)0 on V is positive definite. +(2) for v1, v2 ∈ V , the conditions as follows are satisfied. +([X, v1], v2)0 = −(v1, [Y, v2])0, +([H, v1], v2)0 = −(v1, [H, v2])0. +Theorem 1.1 (Theorem 4.5). g can be decomposed into a direct sum of irre- +ducible s-modules such that s itself is one of the direct summand. All of these +modules except for s are unitarizable. +Also, we classify how many highest weight modules, lowest weight modules, +and modules that are neither highest weight module nor lowest weight module +appear in this decomposition. We regard a root sα1 + tα2 as a point (s, t) in +xy-plane, and We define a region L, −L in xy-plane in §5. If a root α satisfies +α(H) ∈ (0, 2), α ∈ L. If a root α satisfies α(H) ∈ (−2, 0), α ∈ −L. +2 + +Theorem 1.2 (Theorem 7.1). We consider an irreducible decomposition of g +by the action of s. +(1) Let M is an irreducible component of decomposition of g, which contain +a root space for a real root in L. Then, M is an unitary principal or +complementary series representation. +(2) (cf. [Tsu, Proposition 7.3]) There is an unitary principal series represen- +tation containing an 1-dimensional space in h. +(3) g is decomposed into a direct sum of s-submodules described in (1) and (2) +above, s itself, irreducible lowest weight modules, and irreducible highest +weight modules. +We also discuss how to calculate multiplicities of irreducible highest or low- +est modules (§7). Furthermore, we classified irreducible components which are +neither highest weight modules nor lowest weight modules, as either unitary +principal or complementary series representations. +Theorem 1.3 (Theorem 8.11). We consider irreducible components which are +neither highest weight modules nor lowest weight modules and contain root +vectors about real roots in L, obtained by Theorem 7.1. The irreducible com- +ponents are complementary series representations, except those described in +Lemma 8.5 and Lemma 8.10. For the exceptions, the irreducible components +are unitary principal series representations. +2 +General theory of Kac-Moody Lie algebras +Let g be a symmetrizable Kac-Moody Lie algebra on C. Let A be the Cartan +matrix of g and let A be an n × n matrix. Let h be a Cartan subalgebra of g. +Let the Chevalley generators of g be ei, fi, hi, (i = 0, . . . , n − 1). Let hR be the +R-span of Chevalley generators. +From [Kac90, Theorem 2.2], g has a C-valued nondegenerate invariant sym- +metric bilinear form (· | ·) called the standard form. +An antilinear automorphism ω0 of g, called compact involution, is deter- +mined by +ω0(ei) = −fi, +ω0(fi) = −ei +(i = 0, . . . , n − 1), +ω0(h) = −h +(h ∈ hR). +From [Kac90, §2.7], we can determine a nondegenerate Hermitian form (· | ·)0 +on g with (x | y)0 = −(ω0(x) | y). +Write n+ for a subalgebra of g generated by ei’s and n− for a subalgebra of +g generated by fi’s. +We can construct a 3-dimensional subalgebra of g which is spanned by three +non-zero elements J+ ∈ n+, J− ∈ n−, J3 ∈ h. J+, J− and J3 satisfy +[J3, J±] = ±J±, +3 + +[J+, J−] = −J3. +This subalgebra is called SO(1, 2) subalgebra of g. +A representation of SO(1, 2) subalgebra is called unitary if the representation +space V has a Hermitian scalar product (·, ·) and the following two conditions +are satisfied. +(1) The actions of J+ and J− are adjoint each other, and the action of J3 is +self-adjoint. That is, for x, y ∈ V , we have +([J+, x], y) = (x, [J−, y]), +([J3, x], y) = (x, [J3, y]). +(2) Hermitian scalar product (·, ·) is positive definite. +When considering the adjoint action of an SO(1, 2) subalgebra of g to g, from +[Tsu, Lemma 3.1, Lemma 3.2], we can see that the adjoint action satisfying the +condition (1) to be unitary and J− = −ω0(J+) are equivalent. In [NO01], prin- +cipal SO(1, 2) subalgebras for hyperbolic Kac-Moody Lie algebras are studied. +Principal SO(1, 2) subalgebra satisfies that J− = −ω0(J+). +When three non-zero elements X ∈ n+, Y ∈ n−, H ∈ h of g satisfy +[H, X] = 2X, +[H, Y ] = −2Y, +[X, Y ] = H, +these three elements are called sl2-triple of g. A g-subalgebra that these elements +span is called sl2 subalgebra. The SO(1, 2) subalgebras and the sl2 subalgebras +can be converted by +J+ = +1 +√ +2X, +J− = − 1 +√ +2Y, +J3 = 1 +2H. +The condition J− = −ω0(J+) in SO(1, 2) subalgebra is converted to Y = +ω0(X) in sl2 subalgebra. In the following paper, we consider sl2 subalgebra +that satisfies Y = ω0(X). +4 + +3 +sl2-triples of rank 2 hyperbolic symmetric Lie +algebra that is compatible to compact involu- +tion +Let a be an integer that satisfies a ≥ 3, and let g be a hyperbolic Kac-Moody +Lie algebra on C such that the Cartan matrix of g is +� +2 +−a +−a +2 +� +. +Let α0, α1 be the simple roots of g. +Let {Fn} be the sequence of numbers +determined by F0 = 0, F1 = 1, Fk+2 = aFk+1 − Fk. +Lemma 3.1 ([KM95, Proposition 4.4]). The real positive roots of g are of the +form +α = Fk+1α0 + Fkα1 +or +β = Fkα0 + Fk+1α1. +We distinguish these roots as type α and type β, and we also distinguish +root vectors belonging to each root as type α and type β (cf. [Tsu, §4]). +Let X be an element of the space which real positive root vectors span. Then +X can be written as +X = +� +k +ckEk, +(k ∈ {0, . . ., nX − 1}, ck ∈ C, ck ̸= 0, Ek ∈ gβk, Ek ̸= 0) +where βk (k ∈ {0, . . . , nX − 1}) are distinct real roots and nX is a positive +integer. +We call this nX the length of X. Then the following holds. +Lemma 3.2 ([Tsu, Theorem 5.8]). Let X be an element in the space which +real positive root vectors span. +(1) When the length of X is 1 or more than 3, X, Y = ω0(X), H = [X, Y ] +do not form sl2-triple. +(2) Suppose the length of X is 2 and E0, E1 are real positive root vectors +of different types (in the sense of α-type and β-type). Then, taking the +appropriate c0, c1 ∈ C, X = c0E0 + c1E1, Y = ω0(X), and H = [X, Y ] +form sl2-triple. In particular, c0, c1 can be chosen so that c0, c1 ∈ R. +Lemma 3.3 ([Tsu, Theorem 6.4]). Take ⟨H, X, Y ⟩ in Lemma 3.2, (2). +Let +X = c0E0 +c1E1, where E0 is type α and E1 is type β. From Lemma 3.1, using +integers i, j ≥ 0, we can write E0 ∈ gFi+1α0+Fiα1, E1 ∈ gFjα0+Fj+1α1. If and +only if i = j − 1, j, j + 1, H is dominant. +5 + +4 +Irreducible decomposition of g as an sl2 mod- +ule +In this section, we consider an sl2-subalgebra s = ⟨H, X, Y ⟩ of g, which satisfies +the following conditions. +(1) H ∈ h and H is dominant. +(2) X is in the space which is spanned by positive root vectors. +(3) Y = ω0(X). +We show that g is decomposed to irreducible modules by the action of s. +s-module V is called unitarizable if the following conditions are satisfied. +(1) (· | ·)0 on V is positive definite. +(2) for v1, v2 ∈ V , the conditions as follows are satisfied. +([X, v1], v2)0 = −(v1, [Y, v2])0, +([H, v1], v2)0 = −(v1, [H, v2])0. +When Y = ω0(X), the condition (2) are automatically satisfied. +Therefore, +(· | ·)0 is positive definite on V if and only if V is unitarizable. +First, we put +U = {x ∈ g | ∀y ∈ s (x | y)0 = 0}. +U is closed under the action of s, and g = s ⊕ U. +Lemma 4.1. (· | ·)0 is positive definite on U. +Proof. From [Kac90, Theorem 11.7], (· | ·)0 is positive definite on n+ ⊕ n−. The +sign of (· | ·)0 on h is (n − 1, 1). Since s itself is not unitarizable, when we write +h = s ⊕ h′, (· | ·)0 is not positive definite on s. Therefore, (· | ·)0 is positive +definite on h′. Since U = h′ ⊕ n+ ⊕ n−, (· | ·)0 is positive definite on U. +Lemma 4.2. Consider a subspace V of U that is closed under the action of H. +Let V ⊥ be the subspace of U orthogonal to V with respect to the Hermitian +form (· | ·)0. Then U = V ⊕ V ⊥. +Proof. We consider the eigenspace decomposition of U by H. Let Uλ be the +eigenspace for λ and write +U = +� +λ∈C +Uλ. +Since H is a Hermitian operator on (· | ·)0, Uλ and Uµ are orthogonal with +respect to this inner product if λ ̸= µ. +Since H is dominant, Uλ is finite- +dimensional. For each λ, V also inherits the eigenspace decomposition of U. +6 + +Let Vλ be an eigenspace of V for λ, and V can be written as a direct sum of +Vλ’s. Let +V ′ +λ = {v ∈ Vλ | ∀x ∈ Vλ (v | x)0 = 0}, +and +V ′ = +� +λ∈C +V ′ +λ. +Vλ is finite dimensional. From Lemma 4.1, (· | ·)0 is positive definite on U. Thus +we have Uλ = Vλ ⊕ V ′ +λ. Therefore, we have U = V ⊕ V ′ and V ′ = V ⊥. +In the following, we show that U can be decomposed into irreducible modules +by the action of s. +Lemma 4.3. Any non-zero sl2-submodule V of U includes an irreducible sub- +module. +Proof. Take the eigenspace decomposition of U by the action of H. V is also +decomposed into eigenspaces with this decomposition, and each eigenspace of +V is finite-dimensional. We regard H as a linear transform on V and take some +eigenvalue λ of H on V . Let U(sl2) be an universal enveloping algebra of sl2. +Considering the Casimir element C of U(sl2), it preserves Vλ. Since Vλ is finite- +dimensional, there exists an eigenvector of C. Let v denote this. Consider the +sl2-submodule generated by v, which includes an irreducible submodule. +Theorem 4.4. U can be decomposed into direct sum of irreducible s-modules, +and all of these modules are unitarizable. +Proof. We consider a set of irreducible submodules of U such that these sub- +modules are orthogonal to each other with respect to (· | ·)0. Let T be the +set. We order the elements of T by inclusion. Then T is non-empty and in- +ductively ordered. Therefore, from Zorn’s lemma, T has a maximal element. +Take a maximal element of T and denote it by M. Consider the direct sum +of all submodules belonging to M. Let M denote this sum. Suppose U ̸= M, +we derive the contradiction. Since M is a subspace of U which is closed by the +action of H, from Lemma 4.2, we have U = M ⊕ M ⊥. Since M ⊥ is non-zero +sl2 submodule of U, from Lemma 4.3, M ⊥ includes an irreducible submodule. +Let W denote this. we have M ∪ {W} ∈ T , that is contradict the maximality +of M. Therefore, we have U = M, and U can be decomposed into direct sum +of irreducible submodules. Combining this with Lemma 4.1, we can also see the +unitarizability of the modules. +Theorem 4.5. g can be decomposed into direct sum of irreducible s-modules, +which consists s itself. All of these modules except for s are unitarizable. +7 + +5 +sl2 modules in g +In the following, we consider what kind of modules appear when g is decom- +posed into irreducible s-modules. In particular, we consider how many unitary +principal or complementary series representations. +For a lie algebra a, Let U(a) be the universal enveloping algebra of a. Let +V be an irreducible s-module which is an irreducible component of g. +The +Casimir element C of U(s) acts on V by constant multiplication. Let µ be this +constant. From [HT92, Chapter II, Corollary 1.1.11], for an eigenvalue λ0 ∈ C +of H on V , some interval I ⊂ Z exists, and V can be expressed as a direct sum +of 1-dimensional eigenspaces such that the eigenvalues of H are λk = λ0 + 2k +(k ∈ I). From [HT92, Chapter II, Theorem 1.1.13], for an eigenvalue λ of H on +V , we define s1(k) for an integer k as +s1(k) = 8µ − (λ + 2k − 1)2 + 1 +4 +. +(A) +We take an element vk of the eigenspace of V with respect to an eigenvalue +λ + 2k. Then we have X(Y vk) = s1(k)vk. If k ∈ Z such that s1(k) = 0 does +not exist, then V is an irreducible module that is neither highest weight module +nor lowest weight module. If there exists a k ∈ Z such that s1(k) = 0, V is a +highest weight module or a lowest weight module. +Let W be the Weyl group of g. Using Lemma 3.2, we may write H, X, Y in +s as follows. +X = c0w0(ep) + c1w1(eq) +(c0, c1 ∈ R, w0, w1 ∈ W, (p, q) ∈ {(0, 1), (0, 0), (1, 1)}), +Y = −c0w0(fp) − c1w1(fq), +H = −c0w0(hp) − c1w1(hq). +Let ks, ls, ms, ns be real numbers such that c0w0(ep) ∈ gksα0+lsα1, c1w1(eq) ∈ +gmsα0+nsα1. From Lemma 3.3, we can write ks = Fi+1, ls = Fi, ms = Fj, ns = +Fj+1 with integers i, j ≥ 0, and furthermore, i ∈ {j − 1, j, j + 1}. +When we take the root vector E ∈ gsα0+tα1 with s, t ∈ Z, we want to find +out which of the three types of modules E generates under the action of s. +We define L in the xy-plane as follows. L is a region satisfying x ≥ 0, y ≥ +0, (x, y) ̸= (0, 0), x2 − axy + y2 ≤ 1 and the following conditions. +x < ks = Fi+1, +(when i = j − 1) +x + y < ks + ls = Fi + Fi+1, +(when i = j) +y < ls = Fi. +(when i = j + 1) +If we take the root sα0 + tα1 with s, t ∈ Z, then from [KM95, Cor 4.3], the +point in the xy-plane given by (s, t) is in the interior or on the boundary of the +hyperbola x2 − axy + y2 = 1. Let hC be this hyperbola. Let λ ∈ R as the value +for which HE = λE. We have λ = (sα0 + tα1)(H). λ ∈ (0, 2) if and only if +(s, t) ∈ L. In the following, we regard a root sα0 + tα1 as a point (s, t) in the +xy-plane. +8 + +Figure +1: Imaginary roots and real roots in L, a = 3, X = c0r0r1(e0) + +c1r1r0(e1) +0 +2 +4 +6 +8 +0 +2 +4 +6 +8 +x +y +imaginary roots +real roots +Lemma 5.1. We consider the hyperbola hC on the xy-plane. +The hC was +represented by x2 − axy + y2 = 1. Let lb be a line represented by the function +y = −x+ b with some real number b ≥ 0. There are two intersections of hC and +lb. Let p1 and p2 be these points. Let db be a distance between p1 and p2. db is +strictly monotonically increasing with respect to b ≥ 0. The same result holds +when lb is a line represented by y = b or x = b. +Proof. First, we consider the case where lb is represented by y = −x + b. Cal- +culating the y-coordinates of p1, p2 gives +y = (a + 2)b ± +� +(a + 2)(a − 2)b2 + 4(a + 2) +2(a + 2) +. +Therefore, we have +db = +√ +2 · +� +(a + 2)(a − 2)b2 + 4(a + 2) +a + 2 +. +This db is strictly monotonically increasing with respect to b ≥ 0. +Next, we consider the case where lb is represented by y = b. Calculating the +x-coordinates of p1, p2 gives +x = ab ± +� +(a2 − 4)b2 + 4 +2 +. +Therefore, we have +db = +� +(a2 − 4)b2 + 4. +This db is strictly monotonically increasing with respect to b ≥ 0. The same +argument is presented when lb is a line represented by x = b. +9 + +Let R be the interior of hC and hC itself. For s, t ∈ Z, (s, t) is a root if and +only if (s, t) ∈ R. +Lemma 5.2. If (x, y) ∈ L ∪ −L, then neither (x + ks − ms, y + ls − ns) nor +(x − ks + ms, y − ls + ns) are roots. +Proof. First we assume (x, y) ∈ L. The points (ks, ls) and (ms, ns) are on the +hyperbola hC. Let l1 be the line connecting these two points. Using some real +number b > 0, l1 is represented by y = −x+ b when i = j, y = b when i = j − 1, +and x = b when i = j + 1. Let l2 be a line parallel to l1 and passing through +(x, y). Using some real number 0 < b′ < b, l2 is represented by y = −x + b′ +when i = j, y = b′ when i = j − 1, and x = b′ when i = j + 1. Let p11, p12 +be intersections of hC and l1. Let d1 be the distance between p11 and p12. Let +p21, p22 be intersections of hC and l2. Let d2 be the distance between p21 and p22. +From Lemma 5.1, we have d1 > d2. The distance between (ks, ls) and (ms, ns) +is d1. The distance between (x, y) and (x+ks −ms, y+ls −ns) is also d1. These +two points are on l2. The length of the part of l2 that is inside the hyperbola +is d2 < d1. From the fact that (x, y) is inside hC, (x + ks − ms, y + ls − ns) is +outside the hyperbola. Therefore, (x + ks − ms, y + ls − ns) is not in R. The +same argument for (x − ks + ms, y − ls + ns) shows that it is not in R. From +symmetry, the case when (x, y) ∈ −L is also shown. +Lemma 5.3. For a point (s, t) ∈ L corresponding to the root, we consider the +root vector E ∈ gsα0+tα1. Then [X, [Y, E]] ∈ gsα0+tα1. +Proof. We have X = c0w0(ep) + c1w1(eq), Y = −c0w0(fp) − c1w1(fq). Also we +have c0w0(ep) ∈ gksα0+lsα1, c1w1(eq) ∈ gmsα0+nsα1. Then we have +[X, [Y, E]] ∈ gsα0+tα1 + g(s−ks+ms)α0+(t−ls+ns)α1 + g(s−ms+ks)α0+(t−ns+ls)α1. +Since (s, t) is a root, from Lemma 5.2, (s − ks + ms, t − ls + ns) and (s − ms + +ks, t − ns + ls) are not roots. Therefore, we have g(s−ks+ms)α0+(t−ls+ns)α1 + +g(s−ms+ks)α0+(t−ns+ls)α1 = 0, and [X, [Y, E]] ∈ gsα0+tα1. +We consider the Casimir element C of U(s). We can write C = 1 +8H2 − 1 +4H + +1 +2XY . +Lemma 5.4. C acts on a root space as endomorphism. The action is diago- +nalizable. +Proof. From Lemma 5.3, C acts on the root spaces as endomorphism. Since g +is completely reducible as an s-modules, the action on the root space is diago- +nalizable. +Lemma 5.5. For a point (s, t) ∈ L corresponding to the root, we can take the +root vector E ∈ gsα0+tα1 such that E is an eigenvector of the Casimir element +C, and E generates an irreducible s-module. +Proof. From Lemma 5.4, we have the lemma. +10 + +From Lemma 5.5, if we decompose g by the action of s, the decomposition +is compatible with the root space decomposition in the root in L. +We consider how many unitary principal or complementary series represen- +tations appear in the decomposition of g. Since the set of eigenvalues of unitary +principal or complementary series representations is {λ + 2k | k ∈ Z} for some +λ, such a module must contain an eigenspace such that its eigenvalue lie on +[0, 2). Therefore, we consider the root vector of H such that the eigenvalue λ of +H satisfies λ ∈ [0, 2). +If λ = 0, i.e., s = t = 0, Since the dimension of h is 2, there are two +irreducible components of V which have 0-eigenspace (cf. [Tsu, §7]). Since one +is sl2 itself, we consider the other module. The casimir element C acts on this +module by a constant multiple (let µ times). If k satisfies s1(k) = 0, we get +8µ+ 1 = (2k − 1)2. Since µ < −1 from [Tsu, Proposition 7.3], the left hand side +is less than 0. Therefore, there is no integral solution to s1(k) = 0, and this is +an irreducible module that is neither highest weight module nor lowest weight +module. In particular, this module is an unitary principal series representation. +In the following, we consider the case of λ ∈ (0, 2). In this case, (s, t) is a +root in L. We compute [X, [Y, E]]. Since Y = −c0w0(fp) − c1w1(fq), we have +[Y, E] = [−c0w0(fp), E] + [−c1w1(fq), E]. +We have also +[−c0w0(fp), E] ∈ g(s−ks)α0+(t−ls)α1, +[−c1w1(fq), E] ∈ g(s−ms)α0+(t−ns)α1. +If [−c0w0(fp), E] and [−c1w1(fq), E] are not 0, then the eigenvalue of H for them +must be in the (−2, 0) interval. we consider root vectors which the eigenvalue +of H are in the (−2, 0). Since R = −R, the roots with respect to these root +vectors are −L. From Lemma 5.2, if we take two points such that the difference +is (ks − ms, ls − ns) and one of which is a root in −L, then the other is not a +root. Now we have ((s − ms) − (s − ks), (t − ns) − (t − ls)) = (ks − ms, ls − ns). +Therefore, we know that at least one of [−c0w0(fp), E], [−c1w1(fq), E] is zero. +When both of these are 0, we have [Y, E] = 0 and from the fact that C = +1 +8H2 − 1 +4H + 1 +2XY , we can write 8µ = λ2 − 2λ. +When [−c0w0(fp), E] ̸= 0, i.e., (s − ks, t − ls) ∈ R, we have +[X, [Y, E]] = [c0w0(ep), [−c0w0(fp), E]] += [E, [−c0w0(fp), c0w0(ep)]] + [−c0w0(fp), [c0w0(ep), E]]. +We define ps ∈ C by [−c0w0(fp), [c0w0(ep), E]] = psE, then we have +[X, [Y, E]] = [E, c2 +0w0(hp)] + psE. +When ps = 0, we have +[X, [Y, E]] = −[c2 +0w0(hp), E]. +11 + +Therefore in this case, if we let −[c2 +0w0(hp), E] = k0E, then we have 8µ = +λ2 − 2λ + 4k0. +When [c0w0(ep), E] = 0, i.e., (s + ks, t + ls) ̸∈ R, we have ps = 0. +To summarize the above, we take an irreducible decomposition of g by s. let +sα0 +tα1 be a root in L. Let E ∈ gsα0+tα1 such that E generates an irreducible +component of g. Let C be the Casimir element of U(s), and Let µ be a complex +number such that CE = µE. Let k0 and ps be complex numbers satisfying +[−c2 +0w0(hp), E] = k0E, +[−c0w0(fp), [c0w0(ep), E]] = psE. +If (s − ms, t − ns) ̸∈ R, we have +8µ = + + + + + +λ2 − 2λ +((s − ks, t − ls) ̸∈ R) , +λ2 − 2λ + 4k0 +((s − ks, t − ls) ∈ R and (s + ks, t + ls) ̸∈ R) , +λ2 − 2λ + 4k0 + ps +((s − ks, t − ls) ∈ R and (s + ks, t + ls) ∈ R) . +If (s − ks, t − ls) ̸∈ R and not necessarily (s − ms, t − ns) ̸∈ R, we have +8µ = + + + + + +λ2 − 2λ +((s − ms, t − ns) ̸∈ R) , +λ2 − 2λ + 4k0 +((s − ms, t − ns) ∈ R and (s + ms, t + ns) ̸∈ R) , +λ2 − 2λ + 4k0 + ps +((s − ms, t − ns) ∈ R and (s + ms, t + ns) ∈ R) . +Solving +s1(k) = 8µ − (λ + 2k − 1)2 + 1 +4 += 0 +for k on R, we obtain that +k = + + + + + + + + + + + + + + + + + + + + + + + + + +0, 1 − λ +((s − ks, t − ls) ̸∈ R and (s − ms, t − ns) ̸∈ R) , +1 − λ ± +� +(λ − 1)2 + 4k0 +2 + + + +(s − ks, t − ls) ∈ R and (s + ks, t + ls) ̸∈ R +or +(s − ms, t − ns) ∈ R and (s + ms, t + ns) ̸∈ R + + + , +1 − λ ± +� +(λ − 1)2 + 4k0 + ps +2 + + + +(s − ks, t − ls) ∈ R and (s + ks, t + ls) ∈ R +or +(s − ms, t − ns) ∈ R and (s + ms, t + ns) ∈ R + + + . +When (s − ks, t − ls) ̸∈ R and (s − ms, t − ns) ̸∈ R, since (s, t) ∈ L, we have +1 − λ ∈ (−1, 1). Therefore, we know that the only integral solution of s1(k) = 0 +is 0. In this case E belongs to an irreducible lowest weight module. +6 +Classification by roots +Based on the previous section, we classify the root (s, t) in L. We define the +types of roots as follows. +12 + +(1) We say that (s, t) is of type A when (s−ks, t−ls) ̸∈ R and (s−ms, t−ns) ̸∈ +R. +(2) We say that (s, t) is of type B when + + + + + +(s − ks, t − ls) ∈ R and (s + ks, t + ls) ̸∈ R +or +(s − ms, t − ns) ∈ R and (s + ms, t + ns) ̸∈ R + + + + + +. +(3) We say that (s, t) is of type C when + + + + + +(s − ks, t − ls) ∈ R and (s + ks, t + ls) ∈ R +or +(s − ms, t − ns) ∈ R and (s + ms, t + ns) ∈ R + + + + + +. +All roots belong to one of the above types. We put f(x, y) = x2 − axy + y2 for +x, y ∈ R. From [KM95, Cor 4.3], for s, t ∈ Z, (s, t) ̸= (0, 0), (s, t) is a real root if +and only if f(s, t) = 1, and (s, t) is an imaginary root if and only if f(s, t) < 1. +Lemma 6.1. For x, y, x′, y′ ∈ R, if there exists w ∈ W such that (x′, y′) = +w(x, y), then f(x′, y′) = f(x, y). +Proof. It is sufficient to check the case w = r0 and the case w = r1. From the +symmetry, it is sufficient to check the case w = r0. In this case, from the fact +that x′ = ay − x and y′ = y, we have +f(x′, y′) = f(ay − x, y) += (ay − x)2 − ay(ay − x) + y2 += x2 − axy + y2 += f(x, y). +First, we know the following results on real roots. +Lemma 6.2. If (s, t) is a real root in L and s > t, then f(s − ks, t − ls) ≤ 0. +Also, If (s, t) is a real root in L and s < t, then f(s − ms, t − ns) ≤ 0. +Proof. From symmetry, it is sufficient to show f(s − ks, t − ls) ≤ 0 when s > t. +We can write s = Fc+1, t = Fc with c ≥ 0 being an integer. Since ks = Fi+1 +and ls = Fi, we have c < i. Let dic = i − c. From Lemma 6.1, by acting r0 and +r1 on (s − ks, t − ls), we know that +f(s − ks, t − ls) = f(Fc+1 − Fi+1, Fc − Fi) += f(r0(Fc+1 − Fi+1, Fc − Fi)) += f(Fc−1 − Fi−1, Fc − Fi) += f(r1(Fc−1 − Fi−1, Fc − Fi)) += f(Fc−1 − Fi−1, Fc−2 − Fi−2) += · · · +13 + += +� +f(F1 − Fdic+1, F0 − Fdic) +(when c is even) +f(F0 − Fdic, F1 − Fdic+1) +(when c is odd) += f(F1 − Fdic+1, F0 − Fdic). +Since F1 = 1, F0 = 0, we have +f(s − ks, t − ls) = f(1 − Fdic+1, −Fdic) += 2 − aFdic + 2Fdic−1 +< 2 − 2(Fdic − Fdic−1) +≤ 0. +Lemma 6.3. If (s, t) is a real root in L, then (s, t) is of type B. +Proof. First we show that (s, t) is not of type A. From the fact that (s, t) is +a real root and from symmetry, we can write s = Fc+1, t = Fc with c ≥ 0 +being an integer. From ks = Fi+1, ls = Fi, we have c < i. From Lemma 6.2, +f(s − ks, t − ls) ≤ 0. Therefore, (s − ks, t − ls) ∈ R and so we know that (s, t) +is not of type A. +Next, we show that (s, t) is of type B. To show this, we need to show that +(s + ks, t + ls) ̸∈ R. We show f(s + ks, t + ls) > 1. Let dic = i − c. From +Lemma 6.1, by acting r0 and r1 on (s + ks, t + ls), we have +f(s + ks, t + ls) = f(Fc+1 + Fi+1, Fc + Fi) += f(r0(Fc+1 + Fi+1, Fc + Fi)) += f(Fc−1 + Fi−1, Fc + Fi) += f(r1(Fc−1 + Fi−1, Fc + Fi)) += f(Fc−1 + Fi−1, Fc−2 + Fi−2) += · · · += +� +f(F1 + Fdic+1, F0 + Fdic) +(when c is even) +f(F0 + Fdic, F1 + Fdic+1) +(when c is odd) += f(F1 + Fdic+1, F0 + Fdic) += f(1 + Fdic+1, Fdic) += 2 + aFdic − 2Fdic−1 +> 2 + 2(Fdic − Fdic−1) +> 4. +This shows that (s, t) is of type B. +We classify also for imaginary roots in L. +Lemma 6.4. If (s, t), (s′, t′) are imaginary roots, then (s + s′, t + t′) is also +imaginary root. +14 + +Proof. Since f(s, t) ≤ 0, for any r ∈ R, we have f(rs, rt) = r2f(s, t) ≤ 0. It +shows that the line connecting the origin and (s, t) is inside the asymptotes of +the hyperbola x2 − axy + y2 = 1. Similarly, the line connecting the origin and +(s′, t′) is also inside the asymptotes. +Since (s+ s′, t+ t′) is the midpoint of (2s, 2t) and (2s′, 2t′), this point is also +inside the asymptotes. Therefore, (s + s′, t + t′) is an imaginary root. +Lemma 6.5. Let (u, v) ∈ L (u > v) be a real root such that (uα0+vα1)(H) ̸= 0. +Put (s, t) = (ks−u, ls−v). Then (s, t) is a type C imaginary root in L. Similarly, +let (u′, v′) ∈ L (u′ < v′) be a real root such that (u′α0 + v′α1)(H) ̸= 0. Put +(s′, t′) = (ms − u′, ns − v′). Then (s′, t′) ∈ L and (s′, t′) is the imaginary root +of type C. The other imaginary roots in L are of type A. +Proof. From Lemma 6.2, f(−s, −t) = f(s, t) ≤ 0. +It shows that (s, t) is a +imaginary root. We also see that the eigenvalue of H for (s, t) is in the range +(0, 2). Therefore, (s, t) ∈ L is shown. +We show that (s, t) is of type C. To show this, we show that f(s + ks, t + +ls) ≤ 1. Using c ∈ Z, we can write (u, v) = (Fc+1, Fc). Together this with +s + ks = 2ks − u, t + ls = 2ls − v, we have +f(s + ks, t + ls) = f(2Fi+1 − Fc+1, 2Fi − Fc). +Let dic = i−c > 0. From Lemma 6.1, acting r0, r1 on (s+ks, t+ls), i−c = λ ≥ 1, +we have +f(2Fi+1 − Fc+1, 2Fi − Fc) = f(r0(2Fi+1 − Fc+1, 2Fi − Fc)) += f(2Fi−1 − Fc−1, 2Fi − Fc) += f(r1(2Fi−1 − Fc−1, 2Fi − Fc)) += f(2Fi−1 − Fc−1, 2Fi−2 − Fc−2) += · · · += +� +f(2Fdic+1 − F1, 2Fdic − F0) +(when c is even) +f(2Fdic − F0, 2Fdic+1 − F1) +(when c is odd) += f(2Fdic+1 − F1, 2Fdic − F0) += f(2Fdic+1 − 1, 2Fdic) += −2aFdic + 4Fdic−1 + 5 +< −6Fdic + 4Fdic−1 + 5 += (−4Fdic + 4Fdic−1) + (−2Fdic + 5) +< −4 − 2Fdic + 5 +≤ −1. +This shows that f(s+ ks, t+ ls) ≤ −1 and that (s, t) is type C. From symmetry, +we also know that (s′, t′) is in L and is the imaginary root of type C. +Finally, we show the other imaginary roots in L are of type A. Let (s′′, t′′) ∈ L +be such an imaginary root. We show (s′′−ms, t′′−ns) ̸∈ R and (s′′−ks, t′′−ls) ̸∈ +15 + +R. If (s′′ − ms, t′′ − ns) ∈ R or (s′′ − ks, t′′ − ls) ∈ R, (s′′ − ms, t′′ − ns) ∈ −L or +(s′′−ks, t′′−ls) ∈ −L. Since (s′′−ms, t′′−ns)−(s′′−ks, t′′−ls) = (ks −ms, ls− +ns), from Lemma 5.2, we know (s′′ − ms, t′′ − ns) ̸∈ R or (s′′ − ks, t′′ − ls) ̸∈ R. +From symmetry, it is sufficient to consider when (s′′ − ms, t′′ − ns) ̸∈ R. +Under this assumption, (s′′ − ks, t′′ − ls) is an imaginary root or not a root. If +(s′′ − ks, t′′ − ls) is imaginary root, then (ks − s′′, ls − t′′) is also imaginary root +from the symmetry of R. We consider that (ks, ls) = (s′′, t′′) + (ks − s′′, ls − t′′). +The left hand side is real root and the right hand side is the sum of imaginary +roots, which contradicts Lemma 6.4. Therefore, (s′′ − ks, t′′ − ls) is not a root +and (s′′, t′′) is of type A. +The contents of this section can be summarized as follows. +Theorem 6.6. +(1) A real roots in L is of type B. +(2) We consider an imaginary root that can be written as (ks − s, ls − t) or +(ms − s, ns − t) where (s, t) is a real root. Such an imaginary root is of +type C. +(3) The other imaginary roots are of type A. +We now summarize the irreducible s-modules through type A and type C. +For s-modules through type A, we have the following. +Lemma 6.7. An irreducible s-module containing a root vector about a root of +type A in L is a lowest weight module which the root vector is the lowest weight +element. +Proof. Since (s − ks, t − ls) ̸∈ R and (s − ms, t − ns) ̸∈ R for the root (s, t) of +type A, we know that acting Y on the type A root vector will result in 0. This +shows the lemma. +Lemma 6.8. Let M be an irreducible s-module containing a root vector (say +v) with respect to type C root in L. Then one of the following conditions (1), +(2), or (3) is hold. +(1) M is a lowest weight module such that v is a lowest element. +(2) M is a highest weight module such that v is a highest element. +(3) M contains a real root vector with respect to a real root in −L. +Proof. The type C root (s, t) can be written with some real root (sr, tr) that +(ks − sr, ls − tr) or (ms − sr, ns − tr). Therefore, the root vector E of type C +becomes either zero or a real root vector when Y act on it. If E becomes 0 +under the action of Y , then E generates an irreducible lowest weight module. +If E becomes a real root vector, then the real root for this vector is in −L, and +this lemma is shown. +16 + +We also give the type A, B, C distinction to the root of −L by defining +Theorem 6.6. +Then, if there is a unitary principal or complementary series +representation that passes through a root vector of type C in L, −L, it will also +pass through the root vector of type B in −L, L. Therefore, We have only to +classify the modules that contains a type B root space. +Figure 2: a = 3, X = c0r0(e1) + c1r1(e0) +0 +0.5 +1 +1.5 +2 +2.5 +3 +0 +1 +2 +3 +x +y +type A +type B +type C +roots of X +Figure 3: a = 3, X = c0r0(e1) + c1r1r0(e1) +0 +0.5 +1 +1.5 +2 +2.5 +3 +0 +2 +4 +6 +8 +x +y +type A +type B +type C +roots of X +17 + +7 +Irreducible modules which contains a root space +with respect to a type B root +We consider an irreducible decomposition of g by s, and we consider an irre- +ducible component M containing a type B root space. The multiplicity of a real +root space is 1. We can take 0 < λ < 2 such that {λ + 2k′ | k′ ∈ Z} is the set +of the eigenvalues of H in M. We consider the H eigenspace of M such that +the eigenvalue is λ. We assume this eigenspace is gsα0+tα1 such that (s, t) ∈ L, +and (s, t) is real root. We consider k such that s1(k) = 0 in (A) in §5. We show +that it is not an integer. +Let e0, e1, f0, f1, h0 and h1 be Chevalley generators. Using some c0, c1 ∈ +R, w0, w1 ∈ W, and (p, q) ∈ {(0, 1), (0, 0), (1, 1)}, let X = c0w0(ep) + c1w1(eq). +Suppose s > t. We take the root vector E with respect to the root sα0+tα1. We +define λ by HE = λE, and define k0 by [−c2 +0w0(hp), E] = k0E. Thus s1(k) = 0 +implies +k = 1 − λ ± +� +(λ − 1)2 + 4k0 +2 +. +We put +k+ = 1 − λ + +� +(λ − 1)2 + 4k0 +2 +, +k− = 1 − λ − +� +(λ − 1)2 + 4k0 +2 +and we show that k± ̸∈ R or 0 < k± < 1. +When (λ − 1)2 + 4k0 < 0 or (λ − 1)2 + 4k0 ̸∈ R, k± are imaginary numbers. +Therefore we can assume (λ − 1)2 + 4k0 ≥ 0. From 0 < λ < 1, it is clear that +k+ > 0 and k− < 1. To show k+ < 1, we need to show +1 − λ + +� +(λ − 1)2 + 4k0 < 2. +we can easily show that it is reduced to k0 < λ. Also, to show that k− > 0, we +need to show +1 − λ − +� +(λ − 1)2 + 4k0 > 0. +we can easily show that it is reduced to k0 < 0. In summary, we have only to +show that k0 < 0. +First, consider the case (s, t) = (1, 0), i.e., E ∈ gα0. +In this case, from +Lemma 3.3, we have c0w0(ep) ∈ gFi+1α0+Fiα1 and i ≥ 1. Since +k0E = [−c2 +0r0r1r0 . . . r1−p(hp), E] += [−c2 +0(Fi+1h0 + Fih1), E] += −c2 +0(2Fi+1 − aFi)E += −c2 +0(Fi+1 + Fi−1)E, +18 + +we have k0 < 0. When (s, t) = (0, 1), we can show that k0 < 0 by replacing i +with j, p with q and making the same argument. +If (s, t) is general and s > t, we can write (s, t) = (Fi′+1, Fi′). Let p′ be 0 or +1, we can write E = r0r1r0 . . . r1−p′(ep′). From this, we have +[−c2 +0w0(hp), E] = −c2 +0[r0r1r0 . . . r1−p(hp), r0r1r0 . . . r1−p′(ep′)] += −c2 +0r0r1r0 . . . r1−p′[rp′r1−p′rp′ . . . r1−p(hp), ep′]. +We consider k0 and c0 when i is replaced by i−i′, and rewrite them as k′ +0 and c′ +0. +Considering (s, t) = (1, 0) or (0, 1) cases, we have [rp′r1−p′rp′ . . . r1−p(hp), ep′] = +− k′ +0 +c′ +0 ep′. That is, k0 = c2 +0 +c′2 +0 k′ +0. Since k′ +0 < 0, we have k0 < 0. When s < t, we can +show that k0 < 0 as well. +From the above, it can be shown that k0 < 0 in any case, i.e., k is not an +integer. From this, we can see the following. +Theorem 7.1. We consider an irreducible decomposition of g by the action of +s. +(1) Let M is an irreducible component of decomposition of g, which contain +a root space for a type B root sα0 + tα1. Then, M is an unitary principal +or complementary series representation. +(2) (cf. [Tsu, Proposition 7.3]) There is an unitary principal series represen- +tation containing an 1-dimensional space in h. +(3) g is decomposed into a direct sum of s-submodules described in (1) and (2) +above, s itself, irreducible lowest weight modules, and irreducible highest +weight modules. +From [KM95, §3], the multiplicity of each root of g is calculated. Using +this, we can find how many modules appear such that the following condition is +satisfied: the modules are highest or lowest modules, and eigenvalues of H for +root vectors with the highest or the lowest roots are certain value. +First, the modules which contain root spaces in L and −L can be seen from +previous contents. Among the positive root spaces not in L, those with the +smallest eigenvalue in H are considered together. Let λH be their eigenvalue +and dH be their dimensions. Suppose pH modules which contain space with +eigenvalue λH that also contain the root spaces already obtained. Then there +are dH−pH lowest weight modules with the root with eigenvalue λH as the lowest +root. The multiplicities of modules can be obtained inductively by replacing λH +with the next smallest eigenvalue of H and performing the same calculation. +Negative root spaces can be classified by the same calculation. +8 +Unitary principal series representation and com- +plementary series representation +In this section, we consider a module (say M) that is neither highest weight +module nor lowest weight module containing a root vector about the root of type +19 + +B. We compute whether the module is a unitary principal series representation +or a complementary series representation. First, we state the following lemma. +Lemma 8.1. If 8µ ≤ −1, then M is a unitary principal series representation. +If 8µ > −1, then M is a complementary series representation. +Proof. From [HT92, §II 1.2], M is isomorphic to U(ν+, ν−). U(ν+, ν−) is a +sl2-module with H eigenvectors {vn | n ∈ Z} as a basis of linear space, such +that +Hvn = (ν+ − ν− + 2j)vn +(n ∈ Z), +e+vn = (ν+ + n)vn+1, +e−vn = (ν− − n)vn−1, +8µ = (ν+ + ν− − 1)2 − 1. +From [HT92, §III Theorem 1.1.3], if ν+ + ν− = 1, U(ν+, ν−) is a unitary +principal series representation. When 8µ ≤ −1, from +λ = ν+ − ν− ∈ R, +8µ = (ν+ + ν− − 1)2 − 1 < −1, +using b ∈ R we can write +ν+ − ν− = λ, +ν+ + ν− = 1 + bi. +(i = +√ +−1) +In this case, we have +ν+ + ν− = λ + 1 +2 ++ b +2i + −λ + 1 +2 +− b +2i += 1. +Therefore, M is a unitary principal series representation. +Consider the case when 8µ > −1. +From [HT92, §III Theorem 1.1.3], if +ν± ∈ R and if ν− − 1 and −ν+ are both contained in the interval (l − 1, l) with +some l ∈ Z, then U(ν+, ν−) is a complementary series representation. From +8µ > −1, we have +ν+ + ν− = 1 ± +� +8µ + 1, +ν+ − ν− = λ. +Therefore, we have +−ν+, ν− − 1 = −λ − 1 ± √8µ + 1 +2 +. +We show that they are in (−1, 0). +20 + +We show first that 0 < λ < 1. Let n, m be integers such that n > m ≥ 0. +We can write +λ = 2(Fm+1 + Fm) +Fn+1 + Fn +. +It is clear that λ > 0. From a ≥ 3, for integer z ≥ 0, we have +Fz+2 = aFz+1 − Fz +> (a − 1)Fz+1 +≥ 2Fz+1. +Hence we have +Fm+1 + Fm +Fn+1 + Fn +< 1 +2. +Therefore, we have λ < 1. We show that +−1 < −λ − 1 + √8µ + 1 +2 +. +From λ < 1, we have −1 < −λ−1 +2 +. Therefore, this inequality is shown. Next we +show +−λ − 1 + √8µ + 1 +2 +< 0. +We have 8µ = λ(λ − 2) + 4k0. From 0 < λ < 1, we have λ(λ − 2) < 0. Also, +since k0 < 0, we have 8µ < 0. Therefore, we have √8µ + 1 < 1. Using 0 < λ +again, we know that +−λ − 1 + √8µ + 1 +2 +< 0. +For +−λ − 1 − √8µ + 1 +2 +< 0, +this is clear from λ > 0. Finally, we show +−1 < −λ − 1 − √8µ + 1 +2 +. +From k0 < 0 and 8µ = λ2 − 2λ + 4k0, we have λ2 − 2λ > 8µ. From this and +λ < 1 we get 1 − λ > √8µ + 1, which can be transformed to +−1 < −λ − 1 − √8µ + 1 +2 +. +From the above, −λ−1±√8µ+1 +2 +are both in (−1, 0). Therefore, M is a comple- +mentary series representation. +21 + +Hereafter, we want to determine when M is complementary series. First, we +consider the case where i = j. we have +8µ = λ2 − 2λ + 4k0, +λ = 2(Fn+1 + Fn) +Fi+1 + Fi +, +k0 = +−2(2Fi+1 − aFi) +a(F 2 +i + F 2 +i+1) − 4FiFi+1 − 2, +(*) +where n is an integer such that i > n ≥ 0. That is, 8µ is determined by i, n, +and a. We show that 8µ is greater than −1 with finite exceptions. +Lemma 8.2. We assume i = j. If we consider 8µ to be a function of n by (*), +8µ is monotonically decreasing with respect to n. +Proof. k0 is independent on n. λ is monotonically increasing with respect to n. +Since 8µ = λ(λ − 2) + 4k0 and 0 < λ < 1, we know that 8µ is monotonically +decreasing with respect to n. +To show that 8µ is greater than −1 with finite exceptions, we need to examine +when n is large. +Lemma 8.3. We assume i = j, n = i − 1. If we consider 8µ to be a function +of i by (*), 8µ is monotonically increasing with respect to i. +Proof. First we write {Fi} explicitly as follows. The real solutions of x2 − ax + +1 = 0 are x = a± +√ +a2−4 +2 +. As α = a− +√ +a2−4 +2 +, β = a+ +√ +a2−4 +2 +, we can write +Fi = βi − αi +β − α . +From n = i − 1, we have +λ = 2(Fi + Fi−1) +Fi+1 + Fi +, +k0 = +−2(2Fi+1 − aFi) +a(F 2 +i + F 2 +i+1) − 4FiFi+1 − 2. +Let t be a real variable. We define the functions Λ and K0 as follows. +Λ(t) = 2(βt − αt + βt−1 − αt−1) +βt+1 − αt+1 + βt − αt +, +K0(t) = +−2(β − α)(2(βt+1 − αt+1) − a(βt − αt)) +a((βt − αt)2 + (βt+1 − αt+1)2) − 4(βt − αt)(βt+1 − αt+1) − 2, +We have λ = Λ(i) and k0 = K0(i). Using these function, we can calculate as +follows. +d +dtΛ = +4 log β(a + 2)(β − α) +(βt+1 − αt+1 + βt − αt)2 +22 + +d +dt(Λ2 − 2Λ) = 8 log β(a + 2)(β − α) +� +(1 − a)βt − (1 − a)αt + 3βt−1 − 3αt−1� +(βt+1 − αt+1 + βt − αt)3 +d +dtK0 = +2(β − α)(a2 − 4) log β +(a2 − 4)2(β2t+1 + α2t+1 − 2)2 +· +� +2β3t+2 + 2α3t+2 − aβ3t+1 − aα3t+1 ++(3a + 4)βt+1 + (3a + 4)αt+1 − (2a + 6)βt − (2a + 6)αt� +d +dt(Λ2 − 2Λ + 4K0) = 8 log β(β − α) · +� +(a + 2) +� +(1 − a)βt − (1 − a)αt + 3βt−1 − 3αt−1� +(βt+1 − αt+1 + βt − αt)3 ++ +1 +(a2 − 4)(β2t+1 + α2t+1 − 2)2 +· +� +2β3t+2 + 2α3t+2 − aβ3t+1 − aα3t+1 ++(3a + 4)βt+1 + (3a + 4)αt+1 − (2a + 6)βt − (2a + 6)αt� +� +Clearing the denominator, we can calculate as follows. +(βt+1 − αt+1 + βt − αt)3(a − 2)(β2t+1 + α2t+1 − 2)2 +8(a + 2) log β(β − α) +· d +dt(Λ2 − 2Λ + 4K0) += (β6t+3 − α6t+3) − (β6t+2 − α6t+2) ++ (a − 2)(1 − a)(β5t+2 − α5t+2) + 3(a − 2)(β5t+1 − α5t+1) ++ 2(β4t+3 − α4t+3) + (a − 4)(β4t+2 − α4t+2) ++ (11 − 2a)(a − 2)(β3t+2 − α3t+2) − (8a + 1)(a − 2)(β3t+1 − α3t+1) ++ (−11a + 5)(β2t+1 − α2t+1) + 17(β2t − α2t) ++ (8a − 14)(a − 2)(βt+1 − αt+1) + (12a − 6)(a − 2)(βt − αt) ++ 14(β − α) +The coefficient on the left hand side is positive. Using the fact that βt − αt +is monotonically increasing, we can calculate that the right hand side is also +positive. This shows that 8µ = (Λ2 − 2Λ + 4K0)(i) is monotonically increasing +with respect to i. +From Lemma 8.3, we consider the case when i = 1, n = 0. +Lemma 8.4. We assume i = j = 1 and n = 0. If we consider 8µ to be a +function of a by (*), 8µ is monotonically increasing with respect to a. +Proof. Under this assumption, we have +8µ = +−4a2 +a3 − 3a − 2. +Differentiating this as a function of the real variable a, from a ≥ 3, we know +that 8µ is monotonically increasing with respect to a. +23 + +Lemma 8.5. When i = j, we consider s-modules of g that are neither a highest +weight module nor a lowest weight module containing a root vector about the +root of type B obtained by Theorem 7.1. The modules are complementary series +representations, except for the following five types. For these exceptions, the +modules are unitary principal series representations. +(a, i, n) = (4, 1, 0), (3, 1, 0), (3, 2, 1), (3, 3, 2), (3, 4, 3) +Proof. We use Lemma 8.6, Lemma 8.3, and Lemma 8.4. +First, when a = 5, i = 1, n = 0, we have 8µ = − 25 +27 > −1. Therefore, when +a ≥ 5, for any i, n, the module for a, i, n is a complementary series representa- +tion. +Next, when a = 4, i = 1, n = 0, we have 8µ = − 32 +25 < −1. Hence the module +for this is a unitary principal series representation. On the other hand, when +a = 4, i = 2, n = 1, we have 8µ > −1. Therefore, when a = 4, the module for +a, i, n is a complementary series representation except when i = 1, n = 0. +Finally, when a = 3, 8µ < −1 when i = 1, 2, 3, 4 and n = i − 1, and in +these four cases the module is a unitary principal series representation. When +n = i − 2 or i = 5, we have 8µ > −1. Therefore, we know that the module is a +complementary series representation in other cases. +From the above, with five exceptions, neither a highest weight module nor +a lowest weight module containing a root vector about the root of type B is a +complementary series representation. +Next, we consider the case i = j − 1 or i = j + 1. From symmetry, it is +sufficient to consider the case i = j − 1. In this case, λ can be written +λ = 2Fn +Fi+1 +with n as an integer such that i ≥ n ≥ 0. On the other hand, for k0, we have +k0 = +−2(2Fi+1 − aFi) +a(F 2 +i + F 2 +i+1) − 4FiFi+1 − 2 +as for i = j. As with i = j, 8µ is determined by i, n and a. The next lemma is +obvious. +Lemma 8.6. We assume i = j − 1. If we consider 8µ to be a function of n by +(*), 8µ is monotonically decreasing with respect to n. +In the following, we consider whether 8µ is monotonically increasing with +respect to i when n = i. In this case, we have +λ = 2Fi +Fi+1 +. +When i = j, n = i − 1, we already know +λ = 2(Fi + Fi−1) +Fi+1 + Fi +. +24 + +We rewrite as +λ1 = 2Fi +Fi+1 +, +λ2 = 2(Fi + Fi−1) +Fi+1 + Fi +. +Let t be a real variable. We define Λ1, Λ2 and K0 as follows. +Λ1(t) = 2(βt − αt + βt−1 − αt−1) +βt+1 − αt+1 + βt − αt +, +Λ2(t) = 2(βt − αt) +βt+1 − αt+1 , +K0(t) = +−2(β − α)(2(βt+1 − αt+1) − a(βt − αt)) +a((βt − αt)2 + (βt+1 − αt+1)2) − 4(βt − αt)(βt+1 − αt+1) − 2. +We have λ1 = Λ1(i), λ2 = Λ2(i), k0 = K0(i), and 8µ = (Λ2 +1 − 2Λ1 + 4K0)(i). +When i = j, 8µ = (Λ2 +2 − 2Λ2 + 4K0)(i). We compare +d +dtΛ1 and +d +dtΛ2. Since +d +dt(Λ2 +1 − 2Λ1 + 4K0) = 2(Λ1 − 1) d +dtΛ1 + 4 d +dtk0 and 0 < Λ1 − 1 < 1, the smaller +the value of +d +dtΛ1, the larger the value of +d +dt(Λ2 +1 − 2Λ1 + 4K0). We know that +d +dt(Λ2 +2−2Λ2+4K0) > 0. If we show d +dtΛ1 > d +dtΛ2, we also know (Λ2 +2−2Λ2+4K0) +is monotonically increasing with respect to t. Therefore, we know 8µ is also +monotonically increasing with respect to i when i = j − 1, n = i. +Lemma 8.7. +d +dtΛ1 > d +dtΛ2. +Proof. we show that +d +dtΛ1 − d +dtΛ2 > 0. we have +d +dtΛ1 − d +dtΛ2 = +4 log β(a + 2)(β − α) +(βt+1 − αt+1 + βt − αt)2 − 4 log β(β − α) +(βt+1 − αt+1)2 . +Calculating this, we have +(βt+1 − αt+1 + βt − αt)2(βt+1 − αt+1)2 +4 log β(β − α) +� d +dtΛ1 − d +dtΛ2 +� +=(β2t+3 + β2t+3) + (β2t+2 + α2t+2) − (β2t+1 + α2t+1) − (β2t + α2t). +The coefficient on the left hand is positive. We can easily calculate to know +that βt + αt is monotonically increasing with respect to t. From this, we know +the right hand side is also positive. Therefore, +d +dtΛ1 > d +dtΛ2. +Lemma 8.8. We assume i = j − 1, n = i. If we consider 8µ to be a function +of i by (*), 8µ is monotonically increasing with respect to i. +Lemma 8.9. We assume i = 0, j = 1, and n = 0. If we consider 8µ to be a +function of a by (*), 8µ is monotonically increasing with respect to a. +25 + +Proof. Under this assumption, +8µ = +−4 +a − 2. +This is monotonically increasing with respect to a ≥ 3. +Lemma 8.10. When i = j −1, We consider s-modules containing a root vector +about the root of type B that are neither highest weight modules nor lowest +weight modules obtained by Theorem 7.1. +The modules are complementary +series representations, except for the following 23 types. For these exceptions, +the modules are unitary principal series representations. +(a, i, n) =(a′, 0, 0) (6 ≤ a′ ≤ 18), +(5, 0, 0), (5, 1, 1), +(4, 0, 0), (4, 1, 1), +(3, 0, 0), (3, 1, 1), (3, 1, 0), (3, 2, 2), (3, 3, 3), (3, 4, 4) +Proof. We use Lemma 8.6, Lemma 8.8, and Lemma 8.9. +First, when a = +18, i = 0, n = 0, 8µ = −1. Therefore, when a ≥ 18, the modules for a, i, n +are complementary series representations except when (a, i, n) = (18, 0, 0). +Then, when 6 ≤ a ≤ 17, i = 0, n = 0, from 8µ = − 4 +3 > −1, the module for +this pair is a unitary principal series representation. On the other hand, when +6 ≤ a ≤ 17, 8µ < −1 except when (a, i, n) = (a, 0, 0), i.e., the module about +a, i, n is a complementary series representation. +When a = 5, if (a, i, n) = (5, 0, 0), (5, 1, 1), then the modules are unitary +principal series representations, and the others are complementary series repre- +sentations. +When a = 4, if (a, i, n) = (4, 0, 0), (4, 1, 1), then the modules are unitary +principal series representations, and the others are complementary series repre- +sentations. +When a = 3, if (a, i, n) = (3, 0, 0), (3, 1, 1), (3, 1, 0), (3, 2, 2), (3, 3, 3), (3, 4, 4), +then the modules are unitary principal series representations, and the others +are complementary series representations. +From the above, 23 unitary principal series representations are obtained, and +the rest are all complementary series representations. +Theorem 8.11. We consider modules obtained by (1) of Theorem 7.1. The +modules are neither highest weight modules nor lowest weight modules and +contain root vectors about roots of type B. The modules are complementary +series representations, except those enumerated by Lemma 8.5 and Lemma 8.10. +For the exceptions, the modules are unitary principal series representations. +Proof. It can be shown from Lemma 8.5 and Lemma 8.10. +26 + +Acknowledgements +I would like to express my appreciation to my supervisor, Prof. Hisayosi Matu- +moto for his thoughtful guidance. +References +[CM93] D. H. Collingwood, W. M. McGovern, Nilpotent Orbits in Semisimple +Lie Algebras, Van Nostrand Reinhold, 1993 +[Dyn57] E. Dynkin, Semisimple subalgebras of simple Lie algebras, American +Mathematical Society Translations: Series 2, 6, 1957, pp. 111–245 +[GOW02] M. R. Gaberdiel, et al., A class of Lorentzian Kac-Moody algebras, +Nuclear Physics B, 645, 2002, pp. 403–437 +[Kac90] V. G. Kac, Infinite dimensional Lie algebras 3rd edition, Cambridge +university press, 1990 +[KM95] S-J. Kang, D. J. Melville, Rank 2 symmetric hyperbolic Kac-Moody +algebras, Nagoya Mathematical Journal, 140, 1995, pp. 41–75 +[Kos59] B. Kostant, The principal three-dimensional subgroup and the Betti +numbers of a complex simple Lie group, American Journal of Mathemat- +ics, 81, 1959, pp. 973–1032 +[HT92] R. Howe, E. C. Tan, Non-Abelian Harmonic Analysis, Springer-Verlag, +1992 +[NO01] H. Nicolai, D. I. Olive, The Principal SO(1, 2) Subalgebra of a Hy- +perbolic Kac Moody Algebra, Letters in Mathematical Physics, 2001, pp. +141–152 +[Tsu] H. Tsurusaki, sl2 triples whose nilpositive elements are in a space which is +spanned by the real root vectors in rank 2 symmetric hyperbolic Kac-Moody +Lie algebras, Publications of the Research Institute for Mathematical Sci- +ences, to appear +27 + diff --git a/49AyT4oBgHgl3EQfpPiQ/content/tmp_files/load_file.txt b/49AyT4oBgHgl3EQfpPiQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8dece7b7250c7ea6ddf3c24f5b92f06501e37304 --- /dev/null +++ b/49AyT4oBgHgl3EQfpPiQ/content/tmp_files/load_file.txt @@ -0,0 +1,859 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf,len=858 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='00522v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='RT] 2 Jan 2023 Irreducible module decompositions of rank 2 symmetric hyperbolic Kac-Moody Lie algebras by sl2 subalgebras which are generalizations of principal sl2 subalgebras TSURUSAKI Hisanori∗ Abstract There exist principal sl2 subalgebras for hyperbolic Kac-Moody Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In the case of rank 2 symmetric hyperbolic Kac-Moody Lie algebras, certain sl2 subalgebras are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' These subalgebras are generalizations of principal sl2 subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We show that the rank 2 symmetric hyperbolic Kac-Moody Lie algebras themselves are irreducibly decomposed under the action of this sl2 subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Furthermore, we classify irreducible components of the decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In particular, we obtain multiplicities of unitary principal series and complementary series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 1 Introduction A nilpotent orbit in a finite dimensional simple Lie algebra g0 is an orbit ob- tained by acting on the nilpotent element x of g0 by inner automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In [Dyn57], these are classified by weighted Dynkin diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From the Jacobson- Morozov theorem, for a nilpotent element x of g0, we can construct a sl2-triple with x as a nilpositive element ([CM93, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' This makes it equiva- lent to classify nilpotent orbits of g0 and to classify sl2 triples in g0 up to inner automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Among the nilpotent orbits of a finite dimensional simple Lie algebra, the one whose dimension as an algebraic variety is maximal is called the principal nilpotent orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Correspondingly, we can construct a principal SO(3) subalgebra that is compatible with compact involution ([Kos59]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Kac-Moody Lie algebras are generalizations of finite-dimensional simple Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' They are classified into three types: finite type, affine type, and indefinite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The finite type Kac-Moody Lie algebras are finite dimensional simple Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Within indefinite Kac-Moody Lie algebra, there is a class called hyperbolic Kac-Moody Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' A hyperbolic Kac-Moody Lie algebra ∗Graduate School of Mathematical Sciences, University of Tokyo , htsu- rusaki1929@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='com 1 is an indefinite type Kac-Moody Lie algebra such that any true subdiagram of its Dynkin diagram is of finite or affine type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' By analogy with the above theory, in [NO01], for a hyperbolic Kac-Moody Lie algebra, its principal SO(1, 2) subalgebra was constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Note that [GOW02] shows that it is possible to construct a principal SO(1, 2) subalgebra for certain indefinite Kac-Moody Lie algebra that is not hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Corresponding to this principal SO(1, 2) subalgebra, we can construct a principal sl2-subalgebra in a hyperbolic Kac-Moody Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In [Tsu], for the rank 2 symmetric hyperbolic Kac-Moody Lie algebras g, the following re- sult is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let the space that the positive real root vectors span be Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we consider sl2 subalgebras whose nilpositive element exists in Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then we can construct certain sl2 subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' These subalgebras are generalizations of principal sl2 subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In this paper, for an sl2 subalgebra of rank 2 symmetric hyperbolic Kac- Moody Lie algebra g constructed in [Tsu], we show g is decomposed into irre- ducible sl2-modules by its action on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We are going to more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let s be an sl2 subalgebra constructed in [Tsu].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let H, X, Y be an sl2 triple and assume that s is spanned by H, X, Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let the Chevalley generators of g be ei, fi, hi, (i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' , n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let hR be the R-span of Chevalley generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [Kac90, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2], g has a C-valued nondegenerate invariant symmetric bilinear form (· | ·) called the standard form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' An antilinear automorphism ω0 of g, called compact involution, is determined by ω0(ei) = −fi, ω0(fi) = −ei (i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' , n − 1), ω0(h) = −h (h ∈ hR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [Kac90, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='7], we can determine a nondegenerate Hermitian form (· | ·)0 on g with (x | y)0 = −(ω0(x) | y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' s-module V is called unitarizable if the following conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (1) (· | ·)0 on V is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) for v1, v2 ∈ V , the conditions as follows are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' ([X, v1], v2)0 = −(v1, [Y, v2])0, ([H, v1], v2)0 = −(v1, [H, v2])0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' g can be decomposed into a direct sum of irre- ducible s-modules such that s itself is one of the direct summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' All of these modules except for s are unitarizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Also, we classify how many highest weight modules, lowest weight modules, and modules that are neither highest weight module nor lowest weight module appear in this decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We regard a root sα1 + tα2 as a point (s, t) in xy-plane, and We define a region L, −L in xy-plane in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If a root α satisfies α(H) ∈ (0, 2), α ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If a root α satisfies α(H) ∈ (−2, 0), α ∈ −L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 2 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2 (Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider an irreducible decomposition of g by the action of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (1) Let M is an irreducible component of decomposition of g, which contain a root space for a real root in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then, M is an unitary principal or complementary series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' [Tsu, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3]) There is an unitary principal series represen- tation containing an 1-dimensional space in h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (3) g is decomposed into a direct sum of s-submodules described in (1) and (2) above, s itself, irreducible lowest weight modules, and irreducible highest weight modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We also discuss how to calculate multiplicities of irreducible highest or low- est modules (§7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Furthermore, we classified irreducible components which are neither highest weight modules nor lowest weight modules, as either unitary principal or complementary series representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3 (Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider irreducible components which are neither highest weight modules nor lowest weight modules and contain root vectors about real roots in L, obtained by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The irreducible com- ponents are complementary series representations, except those described in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5 and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For the exceptions, the irreducible components are unitary principal series representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 2 General theory of Kac-Moody Lie algebras Let g be a symmetrizable Kac-Moody Lie algebra on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let A be the Cartan matrix of g and let A be an n × n matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let h be a Cartan subalgebra of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let the Chevalley generators of g be ei, fi, hi, (i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' , n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let hR be the R-span of Chevalley generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [Kac90, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2], g has a C-valued nondegenerate invariant sym- metric bilinear form (· | ·) called the standard form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' An antilinear automorphism ω0 of g, called compact involution, is deter- mined by ω0(ei) = −fi, ω0(fi) = −ei (i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' , n − 1), ω0(h) = −h (h ∈ hR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [Kac90, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='7], we can determine a nondegenerate Hermitian form (· | ·)0 on g with (x | y)0 = −(ω0(x) | y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Write n+ for a subalgebra of g generated by ei’s and n− for a subalgebra of g generated by fi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We can construct a 3-dimensional subalgebra of g which is spanned by three non-zero elements J+ ∈ n+, J− ∈ n−, J3 ∈ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' J+, J− and J3 satisfy [J3, J±] = ±J±, 3 [J+, J−] = −J3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' This subalgebra is called SO(1, 2) subalgebra of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' A representation of SO(1, 2) subalgebra is called unitary if the representation space V has a Hermitian scalar product (·, ·) and the following two conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (1) The actions of J+ and J− are adjoint each other, and the action of J3 is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' That is, for x, y ∈ V , we have ([J+, x], y) = (x, [J−, y]), ([J3, x], y) = (x, [J3, y]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) Hermitian scalar product (·, ·) is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When considering the adjoint action of an SO(1, 2) subalgebra of g to g, from [Tsu, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2], we can see that the adjoint action satisfying the condition (1) to be unitary and J− = −ω0(J+) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In [NO01], prin- cipal SO(1, 2) subalgebras for hyperbolic Kac-Moody Lie algebras are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Principal SO(1, 2) subalgebra satisfies that J− = −ω0(J+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When three non-zero elements X ∈ n+, Y ∈ n−, H ∈ h of g satisfy [H, X] = 2X, [H, Y ] = −2Y, [X, Y ] = H, these three elements are called sl2-triple of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' A g-subalgebra that these elements span is called sl2 subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The SO(1, 2) subalgebras and the sl2 subalgebras can be converted by J+ = 1 √ 2X, J− = − 1 √ 2Y, J3 = 1 2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The condition J− = −ω0(J+) in SO(1, 2) subalgebra is converted to Y = ω0(X) in sl2 subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In the following paper, we consider sl2 subalgebra that satisfies Y = ω0(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 4 3 sl2-triples of rank 2 hyperbolic symmetric Lie algebra that is compatible to compact involu- tion Let a be an integer that satisfies a ≥ 3, and let g be a hyperbolic Kac-Moody Lie algebra on C such that the Cartan matrix of g is � 2 −a −a 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let α0, α1 be the simple roots of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let {Fn} be the sequence of numbers determined by F0 = 0, F1 = 1, Fk+2 = aFk+1 − Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1 ([KM95, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The real positive roots of g are of the form α = Fk+1α0 + Fkα1 or β = Fkα0 + Fk+1α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We distinguish these roots as type α and type β, and we also distinguish root vectors belonging to each root as type α and type β (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' [Tsu, §4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let X be an element of the space which real positive root vectors span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then X can be written as X = � k ckEk, (k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=', nX − 1}, ck ∈ C, ck ̸= 0, Ek ∈ gβk, Ek ̸= 0) where βk (k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' , nX − 1}) are distinct real roots and nX is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We call this nX the length of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2 ([Tsu, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let X be an element in the space which real positive root vectors span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (1) When the length of X is 1 or more than 3, X, Y = ω0(X), H = [X, Y ] do not form sl2-triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) Suppose the length of X is 2 and E0, E1 are real positive root vectors of different types (in the sense of α-type and β-type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then, taking the appropriate c0, c1 ∈ C, X = c0E0 + c1E1, Y = ω0(X), and H = [X, Y ] form sl2-triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In particular, c0, c1 can be chosen so that c0, c1 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3 ([Tsu, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Take ⟨H, X, Y ⟩ in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2, (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let X = c0E0 +c1E1, where E0 is type α and E1 is type β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1, using integers i, j ≥ 0, we can write E0 ∈ gFi+1α0+Fiα1, E1 ∈ gFjα0+Fj+1α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If and only if i = j − 1, j, j + 1, H is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 5 4 Irreducible decomposition of g as an sl2 mod- ule In this section, we consider an sl2-subalgebra s = ⟨H, X, Y ⟩ of g, which satisfies the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (1) H ∈ h and H is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) X is in the space which is spanned by positive root vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (3) Y = ω0(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We show that g is decomposed to irreducible modules by the action of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' s-module V is called unitarizable if the following conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (1) (· | ·)0 on V is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) for v1, v2 ∈ V , the conditions as follows are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' ([X, v1], v2)0 = −(v1, [Y, v2])0, ([H, v1], v2)0 = −(v1, [H, v2])0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When Y = ω0(X), the condition (2) are automatically satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, (· | ·)0 is positive definite on V if and only if V is unitarizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First, we put U = {x ∈ g | ∀y ∈ s (x | y)0 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' U is closed under the action of s, and g = s ⊕ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (· | ·)0 is positive definite on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [Kac90, Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='7], (· | ·)0 is positive definite on n+ ⊕ n−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The sign of (· | ·)0 on h is (n − 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since s itself is not unitarizable, when we write h = s ⊕ h′, (· | ·)0 is not positive definite on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, (· | ·)0 is positive definite on h′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since U = h′ ⊕ n+ ⊕ n−, (· | ·)0 is positive definite on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Consider a subspace V of U that is closed under the action of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let V ⊥ be the subspace of U orthogonal to V with respect to the Hermitian form (· | ·)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then U = V ⊕ V ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider the eigenspace decomposition of U by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let Uλ be the eigenspace for λ and write U = � λ∈C Uλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since H is a Hermitian operator on (· | ·)0, Uλ and Uµ are orthogonal with respect to this inner product if λ ̸= µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since H is dominant, Uλ is finite- dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For each λ, V also inherits the eigenspace decomposition of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 6 Let Vλ be an eigenspace of V for λ, and V can be written as a direct sum of Vλ’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let V ′ λ = {v ∈ Vλ | ∀x ∈ Vλ (v | x)0 = 0}, and V ′ = � λ∈C V ′ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Vλ is finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1, (· | ·)0 is positive definite on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Thus we have Uλ = Vλ ⊕ V ′ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we have U = V ⊕ V ′ and V ′ = V ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In the following, we show that U can be decomposed into irreducible modules by the action of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Any non-zero sl2-submodule V of U includes an irreducible sub- module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Take the eigenspace decomposition of U by the action of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' V is also decomposed into eigenspaces with this decomposition, and each eigenspace of V is finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We regard H as a linear transform on V and take some eigenvalue λ of H on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let U(sl2) be an universal enveloping algebra of sl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Considering the Casimir element C of U(sl2), it preserves Vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since Vλ is finite- dimensional, there exists an eigenvector of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let v denote this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Consider the sl2-submodule generated by v, which includes an irreducible submodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' U can be decomposed into direct sum of irreducible s-modules, and all of these modules are unitarizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider a set of irreducible submodules of U such that these sub- modules are orthogonal to each other with respect to (· | ·)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let T be the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We order the elements of T by inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then T is non-empty and in- ductively ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, from Zorn’s lemma, T has a maximal element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Take a maximal element of T and denote it by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Consider the direct sum of all submodules belonging to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let M denote this sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Suppose U ̸= M, we derive the contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since M is a subspace of U which is closed by the action of H, from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2, we have U = M ⊕ M ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since M ⊥ is non-zero sl2 submodule of U, from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3, M ⊥ includes an irreducible submodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let W denote this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we have M ∪ {W} ∈ T , that is contradict the maximality of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we have U = M, and U can be decomposed into direct sum of irreducible submodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Combining this with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1, we can also see the unitarizability of the modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' g can be decomposed into direct sum of irreducible s-modules, which consists s itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' All of these modules except for s are unitarizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 7 5 sl2 modules in g In the following, we consider what kind of modules appear when g is decom- posed into irreducible s-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In particular, we consider how many unitary principal or complementary series representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For a lie algebra a, Let U(a) be the universal enveloping algebra of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let V be an irreducible s-module which is an irreducible component of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The Casimir element C of U(s) acts on V by constant multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let µ be this constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [HT92, Chapter II, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='11], for an eigenvalue λ0 ∈ C of H on V , some interval I ⊂ Z exists, and V can be expressed as a direct sum of 1-dimensional eigenspaces such that the eigenvalues of H are λk = λ0 + 2k (k ∈ I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [HT92, Chapter II, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='13], for an eigenvalue λ of H on V , we define s1(k) for an integer k as s1(k) = 8µ − (λ + 2k − 1)2 + 1 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (A) We take an element vk of the eigenspace of V with respect to an eigenvalue λ + 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then we have X(Y vk) = s1(k)vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If k ∈ Z such that s1(k) = 0 does not exist, then V is an irreducible module that is neither highest weight module nor lowest weight module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If there exists a k ∈ Z such that s1(k) = 0, V is a highest weight module or a lowest weight module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let W be the Weyl group of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2, we may write H, X, Y in s as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' X = c0w0(ep) + c1w1(eq) (c0, c1 ∈ R, w0, w1 ∈ W, (p, q) ∈ {(0, 1), (0, 0), (1, 1)}), Y = −c0w0(fp) − c1w1(fq), H = −c0w0(hp) − c1w1(hq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let ks, ls, ms, ns be real numbers such that c0w0(ep) ∈ gksα0+lsα1, c1w1(eq) ∈ gmsα0+nsα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3, we can write ks = Fi+1, ls = Fi, ms = Fj, ns = Fj+1 with integers i, j ≥ 0, and furthermore, i ∈ {j − 1, j, j + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When we take the root vector E ∈ gsα0+tα1 with s, t ∈ Z, we want to find out which of the three types of modules E generates under the action of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We define L in the xy-plane as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' L is a region satisfying x ≥ 0, y ≥ 0, (x, y) ̸= (0, 0), x2 − axy + y2 ≤ 1 and the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' x < ks = Fi+1, (when i = j − 1) x + y < ks + ls = Fi + Fi+1, (when i = j) y < ls = Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (when i = j + 1) If we take the root sα0 + tα1 with s, t ∈ Z, then from [KM95, Cor 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3], the point in the xy-plane given by (s, t) is in the interior or on the boundary of the hyperbola x2 − axy + y2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let hC be this hyperbola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let λ ∈ R as the value for which HE = λE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We have λ = (sα0 + tα1)(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' λ ∈ (0, 2) if and only if (s, t) ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In the following, we regard a root sα0 + tα1 as a point (s, t) in the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 8 Figure 1: Imaginary roots and real roots in L, a = 3, X = c0r0r1(e0) + c1r1r0(e1) 0 2 4 6 8 0 2 4 6 8 x y imaginary roots real roots Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider the hyperbola hC on the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The hC was represented by x2 − axy + y2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let lb be a line represented by the function y = −x+ b with some real number b ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' There are two intersections of hC and lb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let p1 and p2 be these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let db be a distance between p1 and p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' db is strictly monotonically increasing with respect to b ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The same result holds when lb is a line represented by y = b or x = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First, we consider the case where lb is represented by y = −x + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Cal- culating the y-coordinates of p1, p2 gives y = (a + 2)b ± � (a + 2)(a − 2)b2 + 4(a + 2) 2(a + 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we have db = √ 2 · � (a + 2)(a − 2)b2 + 4(a + 2) a + 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' This db is strictly monotonically increasing with respect to b ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Next, we consider the case where lb is represented by y = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Calculating the x-coordinates of p1, p2 gives x = ab ± � (a2 − 4)b2 + 4 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we have db = � (a2 − 4)b2 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' This db is strictly monotonically increasing with respect to b ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The same argument is presented when lb is a line represented by x = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 9 Let R be the interior of hC and hC itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For s, t ∈ Z, (s, t) is a root if and only if (s, t) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If (x, y) ∈ L ∪ −L, then neither (x + ks − ms, y + ls − ns) nor (x − ks + ms, y − ls + ns) are roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First we assume (x, y) ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The points (ks, ls) and (ms, ns) are on the hyperbola hC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let l1 be the line connecting these two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Using some real number b > 0, l1 is represented by y = −x+ b when i = j, y = b when i = j − 1, and x = b when i = j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let l2 be a line parallel to l1 and passing through (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Using some real number 0 < b′ < b, l2 is represented by y = −x + b′ when i = j, y = b′ when i = j − 1, and x = b′ when i = j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let p11, p12 be intersections of hC and l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let d1 be the distance between p11 and p12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let p21, p22 be intersections of hC and l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let d2 be the distance between p21 and p22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1, we have d1 > d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The distance between (ks, ls) and (ms, ns) is d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The distance between (x, y) and (x+ks −ms, y+ls −ns) is also d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' These two points are on l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The length of the part of l2 that is inside the hyperbola is d2 < d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From the fact that (x, y) is inside hC, (x + ks − ms, y + ls − ns) is outside the hyperbola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, (x + ks − ms, y + ls − ns) is not in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The same argument for (x − ks + ms, y − ls + ns) shows that it is not in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From symmetry, the case when (x, y) ∈ −L is also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For a point (s, t) ∈ L corresponding to the root, we consider the root vector E ∈ gsα0+tα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then [X, [Y, E]] ∈ gsα0+tα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We have X = c0w0(ep) + c1w1(eq), Y = −c0w0(fp) − c1w1(fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Also we have c0w0(ep) ∈ gksα0+lsα1, c1w1(eq) ∈ gmsα0+nsα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then we have [X, [Y, E]] ∈ gsα0+tα1 + g(s−ks+ms)α0+(t−ls+ns)α1 + g(s−ms+ks)α0+(t−ns+ls)α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since (s, t) is a root, from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2, (s − ks + ms, t − ls + ns) and (s − ms + ks, t − ns + ls) are not roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we have g(s−ks+ms)α0+(t−ls+ns)α1 + g(s−ms+ks)α0+(t−ns+ls)α1 = 0, and [X, [Y, E]] ∈ gsα0+tα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider the Casimir element C of U(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We can write C = 1 8H2 − 1 4H + 1 2XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' C acts on a root space as endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The action is diago- nalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3, C acts on the root spaces as endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since g is completely reducible as an s-modules, the action on the root space is diago- nalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For a point (s, t) ∈ L corresponding to the root, we can take the root vector E ∈ gsα0+tα1 such that E is an eigenvector of the Casimir element C, and E generates an irreducible s-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4, we have the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 10 From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5, if we decompose g by the action of s, the decomposition is compatible with the root space decomposition in the root in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider how many unitary principal or complementary series represen- tations appear in the decomposition of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since the set of eigenvalues of unitary principal or complementary series representations is {λ + 2k | k ∈ Z} for some λ, such a module must contain an eigenspace such that its eigenvalue lie on [0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we consider the root vector of H such that the eigenvalue λ of H satisfies λ ∈ [0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If λ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=', s = t = 0, Since the dimension of h is 2, there are two irreducible components of V which have 0-eigenspace (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' [Tsu, §7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since one is sl2 itself, we consider the other module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The casimir element C acts on this module by a constant multiple (let µ times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If k satisfies s1(k) = 0, we get 8µ+ 1 = (2k − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since µ < −1 from [Tsu, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3], the left hand side is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, there is no integral solution to s1(k) = 0, and this is an irreducible module that is neither highest weight module nor lowest weight module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In particular, this module is an unitary principal series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In the following, we consider the case of λ ∈ (0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In this case, (s, t) is a root in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We compute [X, [Y, E]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since Y = −c0w0(fp) − c1w1(fq), we have [Y, E] = [−c0w0(fp), E] + [−c1w1(fq), E].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We have also [−c0w0(fp), E] ∈ g(s−ks)α0+(t−ls)α1, [−c1w1(fq), E] ∈ g(s−ms)α0+(t−ns)α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If [−c0w0(fp), E] and [−c1w1(fq), E] are not 0, then the eigenvalue of H for them must be in the (−2, 0) interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we consider root vectors which the eigenvalue of H are in the (−2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since R = −R, the roots with respect to these root vectors are −L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2, if we take two points such that the difference is (ks − ms, ls − ns) and one of which is a root in −L, then the other is not a root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Now we have ((s − ms) − (s − ks), (t − ns) − (t − ls)) = (ks − ms, ls − ns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we know that at least one of [−c0w0(fp), E], [−c1w1(fq), E] is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When both of these are 0, we have [Y, E] = 0 and from the fact that C = 1 8H2 − 1 4H + 1 2XY , we can write 8µ = λ2 − 2λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When [−c0w0(fp), E] ̸= 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=', (s − ks, t − ls) ∈ R, we have [X, [Y, E]] = [c0w0(ep), [−c0w0(fp), E]] = [E, [−c0w0(fp), c0w0(ep)]] + [−c0w0(fp), [c0w0(ep), E]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We define ps ∈ C by [−c0w0(fp), [c0w0(ep), E]] = psE, then we have [X, [Y, E]] = [E, c2 0w0(hp)] + psE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When ps = 0, we have [X, [Y, E]] = −[c2 0w0(hp), E].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 11 Therefore in this case, if we let −[c2 0w0(hp), E] = k0E, then we have 8µ = λ2 − 2λ + 4k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When [c0w0(ep), E] = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=', (s + ks, t + ls) ̸∈ R, we have ps = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' To summarize the above, we take an irreducible decomposition of g by s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' let sα0 +tα1 be a root in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let E ∈ gsα0+tα1 such that E generates an irreducible component of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let C be the Casimir element of U(s), and Let µ be a complex number such that CE = µE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let k0 and ps be complex numbers satisfying [−c2 0w0(hp), E] = k0E, [−c0w0(fp), [c0w0(ep), E]] = psE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If (s − ms, t − ns) ̸∈ R, we have 8µ = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 λ2 − 2λ ((s − ks, t − ls) ̸∈ R) , λ2 − 2λ + 4k0 ((s − ks, t − ls) ∈ R and (s + ks, t + ls) ̸∈ R) , λ2 − 2λ + 4k0 + ps ((s − ks, t − ls) ∈ R and (s + ks, t + ls) ∈ R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If (s − ks, t − ls) ̸∈ R and not necessarily (s − ms, t − ns) ̸∈ R, we have 8µ = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 λ2 − 2λ ((s − ms, t − ns) ̸∈ R) , λ2 − 2λ + 4k0 ((s − ms, t − ns) ∈ R and (s + ms, t + ns) ̸∈ R) , λ2 − 2λ + 4k0 + ps ((s − ms, t − ns) ∈ R and (s + ms, t + ns) ∈ R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Solving s1(k) = 8µ − (λ + 2k − 1)2 + 1 4 = 0 for k on R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we obtain that k = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 1 − λ ((s − ks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t − ls) ̸∈ R and (s − ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t − ns) ̸∈ R) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 1 − λ ± � (λ − 1)2 + 4k0 2 \uf8eb \uf8ec \uf8ed (s − ks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t − ls) ∈ R and (s + ks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t + ls) ̸∈ R or (s − ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t − ns) ∈ R and (s + ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t + ns) ̸∈ R \uf8f6 \uf8f7 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 1 − λ ± � (λ − 1)2 + 4k0 + ps 2 \uf8eb \uf8ec \uf8ed (s − ks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t − ls) ∈ R and (s + ks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t + ls) ∈ R or (s − ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t − ns) ∈ R and (s + ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t + ns) ∈ R \uf8f6 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When (s − ks, t − ls) ̸∈ R and (s − ms, t − ns) ̸∈ R, since (s, t) ∈ L, we have 1 − λ ∈ (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we know that the only integral solution of s1(k) = 0 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In this case E belongs to an irreducible lowest weight module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 6 Classification by roots Based on the previous section, we classify the root (s, t) in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We define the types of roots as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 12 (1) We say that (s, t) is of type A when (s−ks, t−ls) ̸∈ R and (s−ms, t−ns) ̸∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) We say that (s, t) is of type B when \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 (s − ks, t − ls) ∈ R and (s + ks, t + ls) ̸∈ R or (s − ms, t − ns) ∈ R and (s + ms, t + ns) ̸∈ R \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (3) We say that (s, t) is of type C when \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 (s − ks, t − ls) ∈ R and (s + ks, t + ls) ∈ R or (s − ms, t − ns) ∈ R and (s + ms, t + ns) ∈ R \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' All roots belong to one of the above types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We put f(x, y) = x2 − axy + y2 for x, y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [KM95, Cor 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3], for s, t ∈ Z, (s, t) ̸= (0, 0), (s, t) is a real root if and only if f(s, t) = 1, and (s, t) is an imaginary root if and only if f(s, t) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For x, y, x′, y′ ∈ R, if there exists w ∈ W such that (x′, y′) = w(x, y), then f(x′, y′) = f(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' It is sufficient to check the case w = r0 and the case w = r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From the symmetry, it is sufficient to check the case w = r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In this case, from the fact that x′ = ay − x and y′ = y, we have f(x′, y′) = f(ay − x, y) = (ay − x)2 − ay(ay − x) + y2 = x2 − axy + y2 = f(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First, we know the following results on real roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If (s, t) is a real root in L and s > t, then f(s − ks, t − ls) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Also, If (s, t) is a real root in L and s < t, then f(s − ms, t − ns) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From symmetry, it is sufficient to show f(s − ks, t − ls) ≤ 0 when s > t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We can write s = Fc+1, t = Fc with c ≥ 0 being an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since ks = Fi+1 and ls = Fi, we have c < i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let dic = i − c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1, by acting r0 and r1 on (s − ks, t − ls), we know that f(s − ks, t − ls) = f(Fc+1 − Fi+1, Fc − Fi) = f(r0(Fc+1 − Fi+1, Fc − Fi)) = f(Fc−1 − Fi−1, Fc − Fi) = f(r1(Fc−1 − Fi−1, Fc − Fi)) = f(Fc−1 − Fi−1, Fc−2 − Fi−2) = · · · 13 = � f(F1 − Fdic+1, F0 − Fdic) (when c is even) f(F0 − Fdic, F1 − Fdic+1) (when c is odd) = f(F1 − Fdic+1, F0 − Fdic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since F1 = 1, F0 = 0, we have f(s − ks, t − ls) = f(1 − Fdic+1, −Fdic) = 2 − aFdic + 2Fdic−1 < 2 − 2(Fdic − Fdic−1) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If (s, t) is a real root in L, then (s, t) is of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First we show that (s, t) is not of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From the fact that (s, t) is a real root and from symmetry, we can write s = Fc+1, t = Fc with c ≥ 0 being an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From ks = Fi+1, ls = Fi, we have c < i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2, f(s − ks, t − ls) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, (s − ks, t − ls) ∈ R and so we know that (s, t) is not of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Next, we show that (s, t) is of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' To show this, we need to show that (s + ks, t + ls) ̸∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We show f(s + ks, t + ls) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let dic = i − c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1, by acting r0 and r1 on (s + ks, t + ls), we have f(s + ks, t + ls) = f(Fc+1 + Fi+1, Fc + Fi) = f(r0(Fc+1 + Fi+1, Fc + Fi)) = f(Fc−1 + Fi−1, Fc + Fi) = f(r1(Fc−1 + Fi−1, Fc + Fi)) = f(Fc−1 + Fi−1, Fc−2 + Fi−2) = · · · = � f(F1 + Fdic+1, F0 + Fdic) (when c is even) f(F0 + Fdic, F1 + Fdic+1) (when c is odd) = f(F1 + Fdic+1, F0 + Fdic) = f(1 + Fdic+1, Fdic) = 2 + aFdic − 2Fdic−1 > 2 + 2(Fdic − Fdic−1) > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' This shows that (s, t) is of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We classify also for imaginary roots in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If (s, t), (s′, t′) are imaginary roots, then (s + s′, t + t′) is also imaginary root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since f(s, t) ≤ 0, for any r ∈ R, we have f(rs, rt) = r2f(s, t) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' It shows that the line connecting the origin and (s, t) is inside the asymptotes of the hyperbola x2 − axy + y2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Similarly, the line connecting the origin and (s′, t′) is also inside the asymptotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since (s+ s′, t+ t′) is the midpoint of (2s, 2t) and (2s′, 2t′), this point is also inside the asymptotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, (s + s′, t + t′) is an imaginary root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let (u, v) ∈ L (u > v) be a real root such that (uα0+vα1)(H) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Put (s, t) = (ks−u, ls−v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then (s, t) is a type C imaginary root in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Similarly, let (u′, v′) ∈ L (u′ < v′) be a real root such that (u′α0 + v′α1)(H) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Put (s′, t′) = (ms − u′, ns − v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then (s′, t′) ∈ L and (s′, t′) is the imaginary root of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The other imaginary roots in L are of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2, f(−s, −t) = f(s, t) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' It shows that (s, t) is a imaginary root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We also see that the eigenvalue of H for (s, t) is in the range (0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, (s, t) ∈ L is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We show that (s, t) is of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' To show this, we show that f(s + ks, t + ls) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Using c ∈ Z, we can write (u, v) = (Fc+1, Fc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Together this with s + ks = 2ks − u, t + ls = 2ls − v, we have f(s + ks, t + ls) = f(2Fi+1 − Fc+1, 2Fi − Fc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let dic = i−c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1, acting r0, r1 on (s+ks, t+ls), i−c = λ ≥ 1, we have f(2Fi+1 − Fc+1, 2Fi − Fc) = f(r0(2Fi+1 − Fc+1, 2Fi − Fc)) = f(2Fi−1 − Fc−1, 2Fi − Fc) = f(r1(2Fi−1 − Fc−1, 2Fi − Fc)) = f(2Fi−1 − Fc−1, 2Fi−2 − Fc−2) = · · · = � f(2Fdic+1 − F1, 2Fdic − F0) (when c is even) f(2Fdic − F0, 2Fdic+1 − F1) (when c is odd) = f(2Fdic+1 − F1, 2Fdic − F0) = f(2Fdic+1 − 1, 2Fdic) = −2aFdic + 4Fdic−1 + 5 < −6Fdic + 4Fdic−1 + 5 = (−4Fdic + 4Fdic−1) + (−2Fdic + 5) < −4 − 2Fdic + 5 ≤ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' This shows that f(s+ ks, t+ ls) ≤ −1 and that (s, t) is type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From symmetry, we also know that (s′, t′) is in L and is the imaginary root of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Finally, we show the other imaginary roots in L are of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let (s′′, t′′) ∈ L be such an imaginary root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We show (s′′−ms, t′′−ns) ̸∈ R and (s′′−ks, t′′−ls) ̸∈ 15 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If (s′′ − ms, t′′ − ns) ∈ R or (s′′ − ks, t′′ − ls) ∈ R, (s′′ − ms, t′′ − ns) ∈ −L or (s′′−ks, t′′−ls) ∈ −L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since (s′′−ms, t′′−ns)−(s′′−ks, t′′−ls) = (ks −ms, ls− ns), from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2, we know (s′′ − ms, t′′ − ns) ̸∈ R or (s′′ − ks, t′′ − ls) ̸∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From symmetry, it is sufficient to consider when (s′′ − ms, t′′ − ns) ̸∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Under this assumption, (s′′ − ks, t′′ − ls) is an imaginary root or not a root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If (s′′ − ks, t′′ − ls) is imaginary root, then (ks − s′′, ls − t′′) is also imaginary root from the symmetry of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider that (ks, ls) = (s′′, t′′) + (ks − s′′, ls − t′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The left hand side is real root and the right hand side is the sum of imaginary roots, which contradicts Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, (s′′ − ks, t′′ − ls) is not a root and (s′′, t′′) is of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The contents of this section can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (1) A real roots in L is of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) We consider an imaginary root that can be written as (ks − s, ls − t) or (ms − s, ns − t) where (s, t) is a real root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Such an imaginary root is of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (3) The other imaginary roots are of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We now summarize the irreducible s-modules through type A and type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For s-modules through type A, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' An irreducible s-module containing a root vector about a root of type A in L is a lowest weight module which the root vector is the lowest weight element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since (s − ks, t − ls) ̸∈ R and (s − ms, t − ns) ̸∈ R for the root (s, t) of type A, we know that acting Y on the type A root vector will result in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' This shows the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let M be an irreducible s-module containing a root vector (say v) with respect to type C root in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then one of the following conditions (1), (2), or (3) is hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (1) M is a lowest weight module such that v is a lowest element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) M is a highest weight module such that v is a highest element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (3) M contains a real root vector with respect to a real root in −L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The type C root (s, t) can be written with some real root (sr, tr) that (ks − sr, ls − tr) or (ms − sr, ns − tr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, the root vector E of type C becomes either zero or a real root vector when Y act on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If E becomes 0 under the action of Y , then E generates an irreducible lowest weight module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If E becomes a real root vector, then the real root for this vector is in −L, and this lemma is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 16 We also give the type A, B, C distinction to the root of −L by defining Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then, if there is a unitary principal or complementary series representation that passes through a root vector of type C in L, −L, it will also pass through the root vector of type B in −L, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, We have only to classify the modules that contains a type B root space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Figure 2: a = 3, X = c0r0(e1) + c1r1(e0) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5 3 0 1 2 3 x y type A type B type C roots of X Figure 3: a = 3, X = c0r0(e1) + c1r1r0(e1) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5 3 0 2 4 6 8 x y type A type B type C roots of X 17 7 Irreducible modules which contains a root space with respect to a type B root We consider an irreducible decomposition of g by s, and we consider an irre- ducible component M containing a type B root space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The multiplicity of a real root space is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We can take 0 < λ < 2 such that {λ + 2k′ | k′ ∈ Z} is the set of the eigenvalues of H in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider the H eigenspace of M such that the eigenvalue is λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We assume this eigenspace is gsα0+tα1 such that (s, t) ∈ L, and (s, t) is real root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider k such that s1(k) = 0 in (A) in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We show that it is not an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let e0, e1, f0, f1, h0 and h1 be Chevalley generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Using some c0, c1 ∈ R, w0, w1 ∈ W, and (p, q) ∈ {(0, 1), (0, 0), (1, 1)}, let X = c0w0(ep) + c1w1(eq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Suppose s > t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We take the root vector E with respect to the root sα0+tα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We define λ by HE = λE, and define k0 by [−c2 0w0(hp), E] = k0E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Thus s1(k) = 0 implies k = 1 − λ ± � (λ − 1)2 + 4k0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We put k+ = 1 − λ + � (λ − 1)2 + 4k0 2 , k− = 1 − λ − � (λ − 1)2 + 4k0 2 and we show that k± ̸∈ R or 0 < k± < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When (λ − 1)2 + 4k0 < 0 or (λ − 1)2 + 4k0 ̸∈ R, k± are imaginary numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore we can assume (λ − 1)2 + 4k0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From 0 < λ < 1, it is clear that k+ > 0 and k− < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' To show k+ < 1, we need to show 1 − λ + � (λ − 1)2 + 4k0 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we can easily show that it is reduced to k0 < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Also, to show that k− > 0, we need to show 1 − λ − � (λ − 1)2 + 4k0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we can easily show that it is reduced to k0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In summary, we have only to show that k0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First, consider the case (s, t) = (1, 0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=', E ∈ gα0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In this case, from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3, we have c0w0(ep) ∈ gFi+1α0+Fiα1 and i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since k0E = [−c2 0r0r1r0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' r1−p(hp), E] = [−c2 0(Fi+1h0 + Fih1), E] = −c2 0(2Fi+1 − aFi)E = −c2 0(Fi+1 + Fi−1)E, 18 we have k0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When (s, t) = (0, 1), we can show that k0 < 0 by replacing i with j, p with q and making the same argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If (s, t) is general and s > t, we can write (s, t) = (Fi′+1, Fi′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let p′ be 0 or 1, we can write E = r0r1r0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' r1−p′(ep′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From this, we have [−c2 0w0(hp), E] = −c2 0[r0r1r0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' r1−p(hp), r0r1r0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' r1−p′(ep′)] = −c2 0r0r1r0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' r1−p′[rp′r1−p′rp′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' r1−p(hp), ep′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider k0 and c0 when i is replaced by i−i′, and rewrite them as k′ 0 and c′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Considering (s, t) = (1, 0) or (0, 1) cases, we have [rp′r1−p′rp′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' r1−p(hp), ep′] = − k′ 0 c′ 0 ep′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' That is, k0 = c2 0 c′2 0 k′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since k′ 0 < 0, we have k0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When s < t, we can show that k0 < 0 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From the above, it can be shown that k0 < 0 in any case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=', k is not an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From this, we can see the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider an irreducible decomposition of g by the action of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (1) Let M is an irreducible component of decomposition of g, which contain a root space for a type B root sα0 + tα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then, M is an unitary principal or complementary series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' [Tsu, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3]) There is an unitary principal series represen- tation containing an 1-dimensional space in h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (3) g is decomposed into a direct sum of s-submodules described in (1) and (2) above, s itself, irreducible lowest weight modules, and irreducible highest weight modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [KM95, §3], the multiplicity of each root of g is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Using this, we can find how many modules appear such that the following condition is satisfied: the modules are highest or lowest modules, and eigenvalues of H for root vectors with the highest or the lowest roots are certain value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First, the modules which contain root spaces in L and −L can be seen from previous contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Among the positive root spaces not in L, those with the smallest eigenvalue in H are considered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let λH be their eigenvalue and dH be their dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Suppose pH modules which contain space with eigenvalue λH that also contain the root spaces already obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then there are dH−pH lowest weight modules with the root with eigenvalue λH as the lowest root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The multiplicities of modules can be obtained inductively by replacing λH with the next smallest eigenvalue of H and performing the same calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Negative root spaces can be classified by the same calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 8 Unitary principal series representation and com- plementary series representation In this section, we consider a module (say M) that is neither highest weight module nor lowest weight module containing a root vector about the root of type 19 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We compute whether the module is a unitary principal series representation or a complementary series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First, we state the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If 8µ ≤ −1, then M is a unitary principal series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If 8µ > −1, then M is a complementary series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [HT92, §II 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2], M is isomorphic to U(ν+, ν−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' U(ν+, ν−) is a sl2-module with H eigenvectors {vn | n ∈ Z} as a basis of linear space, such that Hvn = (ν+ − ν− + 2j)vn (n ∈ Z), e+vn = (ν+ + n)vn+1, e−vn = (ν− − n)vn−1, 8µ = (ν+ + ν− − 1)2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [HT92, §III Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3], if ν+ + ν− = 1, U(ν+, ν−) is a unitary principal series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When 8µ ≤ −1, from λ = ν+ − ν− ∈ R, 8µ = (ν+ + ν− − 1)2 − 1 < −1, using b ∈ R we can write ν+ − ν− = λ, ν+ + ν− = 1 + bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (i = √ −1) In this case, we have ν+ + ν− = λ + 1 2 + b 2i + −λ + 1 2 − b 2i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, M is a unitary principal series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Consider the case when 8µ > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [HT92, §III Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3], if ν± ∈ R and if ν− − 1 and −ν+ are both contained in the interval (l − 1, l) with some l ∈ Z, then U(ν+, ν−) is a complementary series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From 8µ > −1, we have ν+ + ν− = 1 ± � 8µ + 1, ν+ − ν− = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we have −ν+, ν− − 1 = −λ − 1 ± √8µ + 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We show that they are in (−1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 20 We show first that 0 < λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let n, m be integers such that n > m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We can write λ = 2(Fm+1 + Fm) Fn+1 + Fn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' It is clear that λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From a ≥ 3, for integer z ≥ 0, we have Fz+2 = aFz+1 − Fz > (a − 1)Fz+1 ≥ 2Fz+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Hence we have Fm+1 + Fm Fn+1 + Fn < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we have λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We show that −1 < −λ − 1 + √8µ + 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From λ < 1, we have −1 < −λ−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, this inequality is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Next we show −λ − 1 + √8µ + 1 2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We have 8µ = λ(λ − 2) + 4k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From 0 < λ < 1, we have λ(λ − 2) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Also, since k0 < 0, we have 8µ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we have √8µ + 1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Using 0 < λ again, we know that −λ − 1 + √8µ + 1 2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For −λ − 1 − √8µ + 1 2 < 0, this is clear from λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Finally, we show −1 < −λ − 1 − √8µ + 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From k0 < 0 and 8µ = λ2 − 2λ + 4k0, we have λ2 − 2λ > 8µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From this and λ < 1 we get 1 − λ > √8µ + 1, which can be transformed to −1 < −λ − 1 − √8µ + 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From the above, −λ−1±√8µ+1 2 are both in (−1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, M is a comple- mentary series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 21 Hereafter, we want to determine when M is complementary series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First, we consider the case where i = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we have 8µ = λ2 − 2λ + 4k0, λ = 2(Fn+1 + Fn) Fi+1 + Fi , k0 = −2(2Fi+1 − aFi) a(F 2 i + F 2 i+1) − 4FiFi+1 − 2, (*) where n is an integer such that i > n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' That is, 8µ is determined by i, n, and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We show that 8µ is greater than −1 with finite exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We assume i = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If we consider 8µ to be a function of n by (*), 8µ is monotonically decreasing with respect to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' k0 is independent on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' λ is monotonically increasing with respect to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since 8µ = λ(λ − 2) + 4k0 and 0 < λ < 1, we know that 8µ is monotonically decreasing with respect to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' To show that 8µ is greater than −1 with finite exceptions, we need to examine when n is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We assume i = j, n = i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If we consider 8µ to be a function of i by (*), 8µ is monotonically increasing with respect to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First we write {Fi} explicitly as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The real solutions of x2 − ax + 1 = 0 are x = a± √ a2−4 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' As α = a− √ a2−4 2 , β = a+ √ a2−4 2 , we can write Fi = βi − αi β − α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From n = i − 1, we have λ = 2(Fi + Fi−1) Fi+1 + Fi , k0 = −2(2Fi+1 − aFi) a(F 2 i + F 2 i+1) − 4FiFi+1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let t be a real variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We define the functions Λ and K0 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Λ(t) = 2(βt − αt + βt−1 − αt−1) βt+1 − αt+1 + βt − αt , K0(t) = −2(β − α)(2(βt+1 − αt+1) − a(βt − αt)) a((βt − αt)2 + (βt+1 − αt+1)2) − 4(βt − αt)(βt+1 − αt+1) − 2, We have λ = Λ(i) and k0 = K0(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Using these function, we can calculate as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='dtΛ = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4 log β(a + 2)(β − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='(βt+1 − αt+1 + βt − αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='dt(Λ2 − 2Λ) = 8 log β(a + 2)(β − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='(1 − a)βt − (1 − a)αt + 3βt−1 − 3αt−1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='(βt+1 − αt+1 + βt − αt)3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='dtK0 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2(β − α)(a2 − 4) log β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='(a2 − 4)2(β2t+1 + α2t+1 − 2)2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2β3t+2 + 2α3t+2 − aβ3t+1 − aα3t+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='+(3a + 4)βt+1 + (3a + 4)αt+1 − (2a + 6)βt − (2a + 6)αt� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='dt(Λ2 − 2Λ + 4K0) = 8 log β(β − α) · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='(a + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='(1 − a)βt − (1 − a)αt + 3βt−1 − 3αt−1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='(βt+1 − αt+1 + βt − αt)3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='(a2 − 4)(β2t+1 + α2t+1 − 2)2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2β3t+2 + 2α3t+2 − aβ3t+1 − aα3t+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='+(3a + 4)βt+1 + (3a + 4)αt+1 − (2a + 6)βt − (2a + 6)αt� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='Clearing the denominator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we can calculate as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (βt+1 − αt+1 + βt − αt)3(a − 2)(β2t+1 + α2t+1 − 2)2 8(a + 2) log β(β − α) d dt(Λ2 − 2Λ + 4K0) = (β6t+3 − α6t+3) − (β6t+2 − α6t+2) + (a − 2)(1 − a)(β5t+2 − α5t+2) + 3(a − 2)(β5t+1 − α5t+1) + 2(β4t+3 − α4t+3) + (a − 4)(β4t+2 − α4t+2) + (11 − 2a)(a − 2)(β3t+2 − α3t+2) − (8a + 1)(a − 2)(β3t+1 − α3t+1) + (−11a + 5)(β2t+1 − α2t+1) + 17(β2t − α2t) + (8a − 14)(a − 2)(βt+1 − αt+1) + (12a − 6)(a − 2)(βt − αt) + 14(β − α) The coefficient on the left hand side is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Using the fact that βt − αt is monotonically increasing, we can calculate that the right hand side is also positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' This shows that 8µ = (Λ2 − 2Λ + 4K0)(i) is monotonically increasing with respect to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3, we consider the case when i = 1, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We assume i = j = 1 and n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If we consider 8µ to be a function of a by (*), 8µ is monotonically increasing with respect to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Under this assumption, we have 8µ = −4a2 a3 − 3a − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Differentiating this as a function of the real variable a, from a ≥ 3, we know that 8µ is monotonically increasing with respect to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 23 Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When i = j, we consider s-modules of g that are neither a highest weight module nor a lowest weight module containing a root vector about the root of type B obtained by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The modules are complementary series representations, except for the following five types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For these exceptions, the modules are unitary principal series representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (a, i, n) = (4, 1, 0), (3, 1, 0), (3, 2, 1), (3, 3, 2), (3, 4, 3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We use Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='6, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3, and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First, when a = 5, i = 1, n = 0, we have 8µ = − 25 27 > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, when a ≥ 5, for any i, n, the module for a, i, n is a complementary series representa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Next, when a = 4, i = 1, n = 0, we have 8µ = − 32 25 < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Hence the module for this is a unitary principal series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' On the other hand, when a = 4, i = 2, n = 1, we have 8µ > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, when a = 4, the module for a, i, n is a complementary series representation except when i = 1, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Finally, when a = 3, 8µ < −1 when i = 1, 2, 3, 4 and n = i − 1, and in these four cases the module is a unitary principal series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When n = i − 2 or i = 5, we have 8µ > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we know that the module is a complementary series representation in other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From the above, with five exceptions, neither a highest weight module nor a lowest weight module containing a root vector about the root of type B is a complementary series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Next, we consider the case i = j − 1 or i = j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From symmetry, it is sufficient to consider the case i = j − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In this case, λ can be written λ = 2Fn Fi+1 with n as an integer such that i ≥ n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' On the other hand, for k0, we have k0 = −2(2Fi+1 − aFi) a(F 2 i + F 2 i+1) − 4FiFi+1 − 2 as for i = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' As with i = j, 8µ is determined by i, n and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The next lemma is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We assume i = j − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If we consider 8µ to be a function of n by (*), 8µ is monotonically decreasing with respect to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In the following, we consider whether 8µ is monotonically increasing with respect to i when n = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In this case, we have λ = 2Fi Fi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When i = j, n = i − 1, we already know λ = 2(Fi + Fi−1) Fi+1 + Fi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 24 We rewrite as λ1 = 2Fi Fi+1 , λ2 = 2(Fi + Fi−1) Fi+1 + Fi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let t be a real variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We define Λ1, Λ2 and K0 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Λ1(t) = 2(βt − αt + βt−1 − αt−1) βt+1 − αt+1 + βt − αt , Λ2(t) = 2(βt − αt) βt+1 − αt+1 , K0(t) = −2(β − α)(2(βt+1 − αt+1) − a(βt − αt)) a((βt − αt)2 + (βt+1 − αt+1)2) − 4(βt − αt)(βt+1 − αt+1) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We have λ1 = Λ1(i), λ2 = Λ2(i), k0 = K0(i), and 8µ = (Λ2 1 − 2Λ1 + 4K0)(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When i = j, 8µ = (Λ2 2 − 2Λ2 + 4K0)(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We compare d dtΛ1 and d dtΛ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since d dt(Λ2 1 − 2Λ1 + 4K0) = 2(Λ1 − 1) d dtΛ1 + 4 d dtk0 and 0 < Λ1 − 1 < 1, the smaller the value of d dtΛ1, the larger the value of d dt(Λ2 1 − 2Λ1 + 4K0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We know that d dt(Λ2 2−2Λ2+4K0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If we show d dtΛ1 > d dtΛ2, we also know (Λ2 2−2Λ2+4K0) is monotonically increasing with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we know 8µ is also monotonically increasing with respect to i when i = j − 1, n = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' d dtΛ1 > d dtΛ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we show that d dtΛ1 − d dtΛ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we have d dtΛ1 − d dtΛ2 = 4 log β(a + 2)(β − α) (βt+1 − αt+1 + βt − αt)2 − 4 log β(β − α) (βt+1 − αt+1)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Calculating this, we have (βt+1 − αt+1 + βt − αt)2(βt+1 − αt+1)2 4 log β(β − α) � d dtΛ1 − d dtΛ2 � =(β2t+3 + β2t+3) + (β2t+2 + α2t+2) − (β2t+1 + α2t+1) − (β2t + α2t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The coefficient on the left hand is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We can easily calculate to know that βt + αt is monotonically increasing with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From this, we know the right hand side is also positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, d dtΛ1 > d dtΛ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We assume i = j − 1, n = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If we consider 8µ to be a function of i by (*), 8µ is monotonically increasing with respect to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We assume i = 0, j = 1, and n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If we consider 8µ to be a function of a by (*), 8µ is monotonically increasing with respect to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Under this assumption, 8µ = −4 a − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' This is monotonically increasing with respect to a ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When i = j −1, We consider s-modules containing a root vector about the root of type B that are neither highest weight modules nor lowest weight modules obtained by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The modules are complementary series representations, except for the following 23 types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For these exceptions, the modules are unitary principal series representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (a, i, n) =(a′, 0, 0) (6 ≤ a′ ≤ 18), (5, 0, 0), (5, 1, 1), (4, 0, 0), (4, 1, 1), (3, 0, 0), (3, 1, 1), (3, 1, 0), (3, 2, 2), (3, 3, 3), (3, 4, 4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We use Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='6, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='8, and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First, when a = 18, i = 0, n = 0, 8µ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, when a ≥ 18, the modules for a, i, n are complementary series representations except when (a, i, n) = (18, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then, when 6 ≤ a ≤ 17, i = 0, n = 0, from 8µ = − 4 3 > −1, the module for this pair is a unitary principal series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' On the other hand, when 6 ≤ a ≤ 17, 8µ < −1 except when (a, i, n) = (a, 0, 0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=', the module about a, i, n is a complementary series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When a = 5, if (a, i, n) = (5, 0, 0), (5, 1, 1), then the modules are unitary principal series representations, and the others are complementary series repre- sentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When a = 4, if (a, i, n) = (4, 0, 0), (4, 1, 1), then the modules are unitary principal series representations, and the others are complementary series repre- sentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When a = 3, if (a, i, n) = (3, 0, 0), (3, 1, 1), (3, 1, 0), (3, 2, 2), (3, 3, 3), (3, 4, 4), then the modules are unitary principal series representations, and the others are complementary series representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From the above, 23 unitary principal series representations are obtained, and the rest are all complementary series representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider modules obtained by (1) of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The modules are neither highest weight modules nor lowest weight modules and contain root vectors about roots of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The modules are complementary series representations, except those enumerated by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5 and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For the exceptions, the modules are unitary principal series representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' It can be shown from Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5 and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 26 Acknowledgements I would like to express my appreciation to my supervisor, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Hisayosi Matu- moto for his thoughtful guidance.' metadata={'source': 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Institute for Mathematical Sci- ences, to appear 27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} diff --git a/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf b/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0dcd6d3f140d732f03da55dc4f6d7fbdc94f389e --- /dev/null +++ b/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c8784a5ccf903997b93ff31d4a354619f65ce27b4b22a6cd22037498885f96d6 +size 572450 diff --git a/6tAyT4oBgHgl3EQf2vk-/vector_store/index.faiss b/6tAyT4oBgHgl3EQf2vk-/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..9dc4e30374397d2e9c6c0553be013a4b239ec07d --- /dev/null +++ b/6tAyT4oBgHgl3EQf2vk-/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2b15ad0f0dfe89aad3ca5f23f177bf240d273a65d57256ed4983d0523831f581 +size 3735597 diff --git a/7dE5T4oBgHgl3EQfQA7X/content/tmp_files/2301.05510v1.pdf.txt b/7dE5T4oBgHgl3EQfQA7X/content/tmp_files/2301.05510v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a450bf242fb431924828ba2f8c658c19e6791011 --- /dev/null +++ b/7dE5T4oBgHgl3EQfQA7X/content/tmp_files/2301.05510v1.pdf.txt @@ -0,0 +1,1929 @@ +Application of Causal Inference Techniques to the +Maximum Weight Independent Set Problem +Jianfeng Liu †§ +Sihong Shao‡∗ +Chaorui Zhang§ +January 16, 2023 +Abstract +A powerful technique for solving combinatorial optimization problems is +to reduce the search space without compromising the solution quality by ex- +ploring intrinsic mathematical properties of the problems. For the maximum +weight independent set (MWIS) problem, using an upper bound lemma which +says the weight of any independent set not contained in the MWIS is bounded +from above by the weight of the intersection of its closed neighbor set and the +MWIS, we give two extension theorems — independent set extension theorem +and vertex cover extension theorem. With them at our disposal, two types +of causal inference techniques (CITs) are proposed on the assumption that a +vertex is strongly reducible (included or not included in all MWISs) or re- +ducible (contained or not contained in a MWIS). One is a strongly reducible +state-preserving technique, which extends a strongly reducible vertex into a +vertex set where all vertices have the same strong reducibility. The other, as a +reducible state-preserving technique, extends a reducible vertex into a vertex +set with the same reducibility as that vertex and creates some weighted pack- +ing constraints to narrow the search space. Numerical experiments show that +our CITs can help reduction algorithms find much smaller remaining graphs, +improve the ability of exact algorithms to find the optimal solutions and help +heuristic algorithms produce approximate solutions of better quality. In par- +ticular, detailed tests on 12 representative graphs generated from datasets in +†Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China +‡CAPT, LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, P.R. +China +§Theory Lab, Central Research Institute, 2012 Labs, Huawei Technologies Co., Ltd. +∗Author to whom correspondence should be addressed: sihong@math.pku.edu.cn +1 +arXiv:2301.05510v1 [math.OC] 13 Jan 2023 + +Network Data Repository demonstrate that, compared to the state-of-the-art +algorithms, the size of remaining graphs is further reduced by more than 32.6%, +and the number of solvable instances is increased from 1 to 5. +AMS subject classifications: 05C69; 68W40; 90C06; 90C27; 90C57 +Keywords: maximum weight independent set; independent set extension; +vertex cover extension; causal inference techniques; reduction algorithm; exact +algorithm; heuristic algorithm; Network Data Repository. +1 +Introduction +Let G = (V, E, w) be an undirected vertex-weighted graph, where each vertex v ∈ V +is associated with a weight w(v) ∈ R+. A subset I ⊆ V is called an independent set +if its vertices are pairwise non-adjacent, and the vertex cover of graph G is a subset +of vertices V C ⊆ V such that every edge e ∈ E is incident to at least one vertex +in subset V C. Independent set and vertex cover are two complementary concepts +in graph and can be transformed into each other on demand [29]. The maximum +weight independent set (MWIS) problem is to find the independent set of largest +weight among all possible independent sets and the weight of a MWIS of graph G is +denoted by αw(G), while the minimum weight vertex cover (MWVC) problem asks for +the vertex cover with the minimum weight. Furthermore, if subset I ⊆ V is a MWIS, +then subset V C = V \I is a MWVC, and vice versa [6, 29]. The MWIS problem is an +extension of the maximum independent set (MIS) problem, which is a classic NP-hard +problem [13, 9]. It can be applied to various real-world problems, such as information +retrieval [4], computer vision [12], combinatorial auction problem [29] and dynamic +map labeling problem [17]. +Due to its wide range of practical applications, the +research on efficient algorithms for computing the MWIS is of great significance. +Most previous work are focused on heuristic algorithms to find near-optimal solutions +in reasonable time [24, 20, 6, 18], while exact algorithms, usually referring to Branch- +and-Bound (B&B) methods [3, 26, 2, 22], become infeasible when the size of problem +increases. +Recently, it has been well demonstrated that reduction rules (a.k.a. kerneliza- +tion) are very effective in practice for solving the MIS problem [25]. These rules mine +the structural properties of underlying graph and reduce the search space by such +as removing vertices, contracting subgraphs, restricting the set of independent sets, +etc., to produce a smaller kernel graph such that the MIS of the original graph can be +recovered from the MIS of the kernel. After integrating them, some state-of-the-art +exact solvers are able to solve the MIS problem on many large real networks [11]. +2 + +These solvers can be usually divided into two types: One performs the kernelization +only once and runs the B&B algorithm [23, 16] on the kernelized instance, while +the other joins hands with the Branch-and-Reduce (B&R) algorithm [19] and per- +forms reduction in every branch of the search tree. As for those instances that can’t +be solved exactly, high-quality solutions can be found by combining kernelization +with local search [8, 10]. Moreover, when a vertex is selected for branching in the +branching process of the B&R algorithm, if it is assumed to be in all MISs, then +its satellite set will also be in all MISs [14], while its mirror set will be removed +directly from the graph, if it is assumed not to be in all MISs [13]. Further, a conflict +analysis on the assumption that a vertex is in all MISs can be also plugged in to +find some contradictions and the concept of “unconfined/confined vertices” was in- +troduced [28]. Later, an auxiliary constraint called packing constraint was proposed +to accelerate the B&R algorithm by simply exploring branches that satisfy all pack- +ing constraints [1]. The central idea behind all these attempts for the MIS problem +involves a state-preserving technique which starts from a vertex, named the starting +vertex for convenience, and then finds a vertex set with the same state as the starting +vertex to reduce the search space, thereby implying that some subsequent operations +can be implemented on the resulting vertex set instead of only on the starting vertex. +For the MWIS problem, similar state-preserving techniques are rarely used except +for a recent work using unconfined/confined vertices [27], though some simple and +fast reduction rules have been used in B&R algorithms [15, 27]. To this end, we +devote ourselves into developing state-preserving techniques for the MWIS problem +in this work. The state of the starting vertex we consider can be +• strongly reducible, meaning that the vertex is included in all MWISs/MWVCs; +or +• reducible, meaning that the vertex is contained in a MWIS/MWVC. +Considering that the assumed state of the starting vertex must be used to analyze its +local structure to obtain inference results, these targeted state-preserving techniques +are called causal inference techniques (CITs). Inspired by their success in solving the +MIS problem, we will systematically develop CITs to solve the MWIS problem by +analyzing intrinsic mathematical properties of underlying graph. More specifically, +our main contributions are in three aspects as follows. +First, by virtue of the upper bound lemma, i.e., the weight of any independent set +not contained in the MWIS is bounded from above by the weight of the intersection +of its closed neighbor set with the MWIS, two extension theorems are developed. +With them, we propose a series of CITs which have been rarely used previously in +3 + +the MWIS problem. According to the state of the starting vertex, our CITs can be +divided into two categories. The first type is a strongly reducible state-preserving +technique. We first assume that the starting vertex is strongly reducible, and then +try to extend this vertex to obtain a vertex set with the same strong reducibility. +If the upper bound lemma is not satisfied in this process, then this contradicts +the assumption, and the starting vertex can be removed from the graph directly. +Otherwise, combined with the state-preserving result obtained from the previous +process, we continue to search for a set called the simultaneous set, which is either +included in a MWIS or contained in a MWVC. The second type is a reducible state- +preserving technique. Under the assumption that the starting vertex is reducible, a +vertex set with the same reducibility can be obtained by extending from this vertex. +Moreover, if this vertex is selected for branching in the B&R algorithm, with the +upper bound lemma, an inequality constraint called weight packing constraint will +be created to restrict subsequent searches. +Next, according to the characteristics of the proposed CITs, we integrate them +into the existing algorithmic framework. The first type of CIT can be used to design +reduction rules to simplify graph. +These reduction rules are integrated into the +existing reduction algorithm. In the B&R algorithm, when a vertex is selected to +branch, a vertex set and a weight packing constraint depending on the assumed +state of the vertex can be obtained from state-preserving results of two types of +CITs. The vertex set is used to further simplify the corresponding branch, while we +can prune branches that violate constraints and simplify the graph by maintaining all +created weight packing constraints. During the local search process of the heuristic +algorithm, when the state of a vertex needs to be changed, all vertex states in the +vertex set obtained by the second type of CIT will also be modified to be the same +as that vertex, which expands the area of local search and improves the ability of +local search to find better local optima. +Numerical experiments on 12 representative graphs generated from datasets in +Network Data Repository show that the performance of various algorithms is greatly +improved after integrating our CITs. The size of the kernel obtained by the resulting +reduction algorithm is greatly reduced. In addition, compared to the state-of-the-art +exact algorithm, the number of solvable instances have been increased from 1 to 5. +And the ability of the heuristic algorithm to find better local optimal solutions is +significantly improved. These experimental results form the third major contribution +of this paper. +Relevant notations used in this work are given in Table 1 and the rest of the paper +is organized as follows. We present two extension theorems in Section 2 and detail +CITs in Section 3. How the CITs are combined with existing algorithmic frameworks +4 + +G = (V,E,w) +an undirected vertex-weight graph G with vertex set V , edge set E and vertex weight function w : V → R+ +N(v) = {u ∈ V |{u,v} ∈ E} +the neighbor set of vertex v +N[v] = N(v) ∪ {v} +the closed neighbor set of vertex v +N(S) = ( +� +v∈S +N(v))\S +the open neighbor set of set S +N[S] = N(S) ∪ {S} +the closed neighbor set of set S +|S| +the size of set S +w(S) = +� +v∈S +w(v) +the weight of all vertices in set S +d(v) +the degree of a vertex v +dist(u,v) +the minimum number of edges in the path from vertex u to vertex v +Nl(v) = {u| dist(u,v) = l} +the set of vertices at distance l from vertex v, +in particular, N1(v) = N(v) +G[S] = (S,ES,w), +∀e = {u,v} ∈ ES, u,v ∈ S +the subgraph induced by a non-empty vertex subset S of V +α(G) +the size of a MIS of unweight graph G +αw(G) +the weight of a MWIS of graph G +AI +the set of all MWISs in graph G +AC +the set of all MWVCs in graph G +S ◀ AI +set S is an independent set and is included in all MWISs +C ◁ AC +set C is contained in all MWVCs +vertex v is strongly reducible +vertex v is included in all MWISs/MWVCs +vertex v is reducible +vertex v is contained in a MWIS/MWVC +vertex v is strongly inclusive +vertex v is included in all MWISs +vertex v is strongly sheathed +vertex v is contained in all MWVCs +vertex v is inclusive +vertex v is included in a MWIS +vertex v is sheathed +vertex v is contained in a MWVC +set S is strongly inclusive +set S is an independent set and is included in all MWISs +set S is strongly sheathed +set S is contained in all MWVCs +set S is inclusive +set S is an independent set and is included in a MWIS +set S is sheathed +set S is contained in a MWVC +independent set S is strongly exclusive +independent set S is not contained in all MWIS +independent set S is exclusive +independent set S is not contained in a MWIS +a set S called a simultaneous set +set S is either included in a MWIS or contained in a MWVC +Table 1: Notations used throughout the paper. +is described in Section 4. Extensive numerical tests are carried out in Section 5 to +verify the performance improvement of integrating our CITs into existing algorithmic +frameworks in terms of efficiency and accuracy. The paper is concluded in Section 6 +with a few remarks. +2 +Two Extension Theorems +The theoretical cornerstones of CITs in this paper are two extension theorems: inde- +pendent set extension theorem and vertex cover extension theorem. Before delineat- +ing them, we need to have a deep understanding of the local structure of the MWIS +and first give the upper bound lemma. +Lemma 2.1 (upper bound lemma). Let set IC be an independent set in the graph. +(a) Suppose there is an Iw ∈ AI such that IC ̸⊆ Iw, then w(Iw ∩ N[IC]) ⩾ w(IC) +holds. +(b) Assume that IC ̸⊆ I, ∀I ∈ AI holds, then it satisfies: w(IC) < w(I∩N[IC]), ∀I ∈ +AI. +Proof. Proof We first prove (a) by contradiction. If not, we can obtain an indepen- +dent set I′ +w = (Iw\N[IC])∪(IC) such that w(I′ +w) = w(Iw)+w(IC)−w(Iw ∩N[IC]) > +w(Iw), a contradiction. +Next, we consider (b). If there is an I1 ∈ AI such that w(IC) ⩾ w(I1 ∩ N[IC])), +we can construct an independent set I′ +1 = (I1\N[IC])∪IC satisfying w(I′ +1) = w(I1)+ +w(IC) − w(I1 ∩ N[IC]) ⩾ w(I1). +Then I′ +1 ∈ AI and IC ⊆ I′ +1, which leads to a +contradiction. +5 + +The upper bound lemma describes such a property: For any independent set that +is (strongly) exclusive, the weight of the intersection of its closed neighbor set with +the MWIS is the upper bound on its weight. With it, the independent set extension +theorem can be introduced as follows. +Theorem 2.2 (Independent Set Extension Theorem). Let sets IS and S be two +independent sets in the graph. +(a) Assume that there exists an Iw ∈ AI such that IS ⊆ Iw. +If there is an +independent set IS′ ⊆ N(IS) such that w(IS′) > w(IS ∩ N(IS′)), then +there exists an independent set IS′′ ⊆ N(IS′)\N[IS] satisfying the inequal- +ity: w(IS′) ⩽ w(IS ∩ N(IS′)) + w(IS′′). In addition, IS ∪ IS′′ ⊆ Iw if such +IS′′ is unique. +(b) Suppose S ◀ AI, then for any independent set S′ ⊆ N(S), there is an indepen- +dent set S′′ ⊆ N(S′)\N[S] such that w(S′) < w(S ∩ N(S′)) + w(S′′). Besides, +if such S′′ is unique, then S ∪ S′′ ◀ AI. +Proof. Proof We first consider the proof of (a), and it is obvious that IS′ ̸⊆ Iw. In +view of the fact that the relationship between Iw and N[IS′] satisfies: Iw ∩ N[IS′] = +Iw∩N(IS′) = (IS∩N(IS′))∪(Iw∩(N(IS′)\N[IS])) and by the upper bound lemma, +we can get: w(IS ∩ N(IS′)) + w(Iw ∩ (N(IS′)\N[IS])) = w(Iw ∩ N(IS′)) = w(Iw ∩ +N[IS′]) ⩾ w(IS′). Thus, the existence of such IS′′ is proved. Furthermore, assuming +that such IS′′ is unique, then IS′′ = Iw ∩ (N(IS′)\N[IS]) and IS ∪ IS′′ ⊆ Iw. +Similar ideas can be used to prove (b). Obviously S′ ̸⊆ I, ∀I ∈ AI holds, so from +the upper bound lemma, it can be directly obtained: ∀I ∈ AI, w(I ∩N[S′]) > w(S′). +Further, by considering that the relationship between I and N[S′] satisfies: I ∩ +N[S′] = I ∩N(S′) = (S∩ N(S′))∪(I ∩(N(S′)\N[S])), we prove the existence of such +S′′. Also, if such S′′ is unique, the following result holds: S′′ = I∩(N(S′)\N[S]), ∀I ∈ +AI, and then S ∪ S′′ ◀ AI. +The independent set extension theorem gives a method for extending independent +set that is (strongly) inclusive: Try to find an independent set to add to the extended +independent set, and that independent set is the only one that guarantees that the +upper bound lemma is satisfied in the local structure of the extended independent +set. +Next, with the help of the upper bound lemma, the vertex cover extension +theorem is given below. +Theorem 2.3 (Vertex Cover Extension Theorem). Let sets IC and C be two vertex +subsets in the graph. +6 + +(a) Suppose set IC ⊆ V Cw, then the vertices in IC have the property: ∀p ∈ IC, +w(p) ⩽ αw(G[N(p)\IC]). Also, for a vertex v ∈ IC and a vertex u ∈ N 2(v), +IC ∪ {u} ⊆ V Cw holds if the inequality w(v) > αw(G[N(v)\(IC ∪ N(u))]) is +satisfied. +(b) Assume that set C ◁ AC, then ∀p ∈ C, w(p) < αw(G[N(p)\C]) is always +satisfied. In addition, if there exists a vertex v ∈ C and a vertex u ∈ N 2(v) +such that w(v) ⩾ αw(G[N(v)\(C ∪ N(u))]), then C ∪ {u} ◁ AC. +Proof. Proof We first consider (a) and let set Iw = V \V Cw. From the upper bound +lemma, these results can be directly obtained: ∀p ∈ IC, w(p) ⩽ w(Iw ∩ N[p]) = +w(Iw ∩ N(p)) ⩽ αw(G[N(p)\IC]). Also, based on the assumption about u in (a), +if u ∈ Iw, then w(v) ⩽ w(Iw ∩ N[v]) = w(Iw ∩ N(v)) ⩽ αw(G[N(v)\(IC ∪ N(u))]), +which leads to a contradiction. +Similar methods can be used to prove (b). +First, ∀p ∈ C, ∀I ∈ AI, w(p) < +w(I ∩N[p]) = w(I ∩N(p)) ⩽ αw(G[N(p)\C]) can be obtained from the upper bound +lemma. Besides, under given conditions about u in (b), if there is an I∗ ∈ AI such +that u ∈ I∗, a contradiction is deduced from w(p) < w(I∗ ∩ N[p]) = w(I∗ ∩ N(p)) ⩽ +αw(G[N(p)\(C ∪ N(u))]). +The vertex cover extension theorem describes how to expand a set that is (strongly) +sheathed: Attempt to find a vertex that satisfies the condition that after removing +its neighbor set, the upper bound lemma is not satisfied in the local structure of the +expanded set. If such a vertex is found, it is directly added to the expanded set. +3 +Causal Inference Techniques +In this section, with the help of the upper bound lemma and two extension theorems, +we give the CITs used in this paper. Our CITs can be divided into two types: The +first type is a strongly reducible state-preserving technique introduced in Section 3.1, +while the second type is a reducible state-preserving technique shown in Section 3.2. +3.1 +Strongly reducible state-preserving technique +The strongly reducible state-preserving technique exploits the assumption that a +vertex is strongly reducible, and the assumed state of the vertex can be divided into +two cases: The vertex is assumed to be strongly inclusive or is assumed to be strongly +sheathed. We first consider the assumption that a vertex is strongly inclusive and +give the following definition. +7 + +Definition 3.1. Let set S be an independent set in the graph. If a vertex u ∈ N(S) +such that w(u) ⩾ w(S ∩ N(u)), we call it a child of set S. A child u is called an +extending child if and only if there exists a unique independent set S∗ ⊆ N(u)\N[S] +such that w(u) < w(S ∩ N(u)) + w(S∗) and vertex set S∗ is called a satellite set of +set S. +On the basis of Definition 3.1, with the assumption that a vertex is strongly inclu- +sive, the concept of ‘confined/unconfined vertices’ is given by the following conflict +analysis process: +Definition 3.2. Let v be a vertex in the graph. Suppose set S := {v} ◀ AI, repeating +(i) until (ii) or (iii) holds: +(i) As long as set S has an extending child in N(S), set S is extended by including +the corresponding satellite set into set S. +(ii) If a child u such that w(u) ⩾ w(S ∩N(u))+αw(G[N(u)\N[S]]) could be found, +that is, the upper bound lemma is not satisfied in the local structure of set S, +then halt and vertex v is called an unconfined vertex. +(iii) If any child is not an extending child, then halt and return set Sv = S. In this +case, vertex v is called a confined vertex and the set Sv is called the confining +set of vertex v. +Some examples of unconfined vertex are given in Figure 1. By means of the con- +flict analysis process in Definition 3.2, vertices a and h can be found to be unconfined +vertices. It is also worth noting that, by the definition of unconfined vertex given +in [27], in Figure 1, only vertex a can be found to be an unconfined vertex. The +reason for this is that we further generalize the concept of confined/unconfined ver- +tices in this work. Compared with the definition of extending child u in [27], which +requires |N(u)\N[S]| = 1 and w(u) < w(N(u)\N(S)), we can consider the more +general case where N(u)\N[S] is an independent set rather than a single vertex, +helping us find more unconfined vertices. +Next, we will explore the properties of confined/unconfined vertices. +By the +conflict analysis process in Definition 3.2 and the independent set extension theorem, +set S can be extended under the assumption: set S := {v} ◀ AI, and set S ◀ AI is +always satisfied. If vertex v is a unconfined vertex, then the upper bound lemma is not +satisfied in the local structure of set S, which contradicts set S ◀ AI. Thus, vertex v +is sheathed. Otherwise, then there is a state-preserving result, i.e., the corresponding +confining set Sv ◀ AI holds. Furthermore, suppose two confined vertices u, v and the +corresponding confining sets Su, Sv such that u ∈ Sv and v ∈ Su. If {v} ◀ AI, then +8 + +Figure 1: +Some examples of unconfined vertices, and a MWIS in this graph +is {b, d, g, i}. +Let set S := {a}, from Definition 3.1, vertex b is an extend- +ing child of set S and set {c} is a satellite set of set S. +Thus, set S can be +extended as: +{a, c}. +At this time, it can be found that a child d such that +w(d) ⩾ w(S ∩ N(d)) + αw(G[N(d)\N[S]]), then halt and conclude that vertex a +is an unconfined vertex. +Similarly, let set S := {h}, then it can be found that +vertex g is an extending child of set S and set {f, l} is a satellite set of set S. +So set S can be further expanded as: {h, l, f}. +After that, the child i satisfied: +w(i) ⩾ w(S ∩ N(i)) + αw(G[N(i)\N[S]]), hence, vertex h is an unconfined vertex. +obviously {u} ◀ AI holds. If not, vertex v is sheathed in graph G. Since v ∈ Su, +then vertex v is included in the satellite set of an intermediate state set S′ of Su, +which means that in graph G[V \{v}], the upper bound lemma is not satisfied in the +local structure of set S′. Thus, by Definition 3.2, vertex u is an unconfined vertex +of graph G[V \{v}] and is sheathed in this graph. From these analysis results and +the symmetry of the relationship between vertex v and vertex u, we can know that +vertex set {u, v} is a simultaneous set. Therefore, the following properties can be +obtained: +Corollary 3.3. Let v is a vertex in the graph. +(a) If vertex v is an unconfined vertex, then it is sheathed and after deleting it from +the graph, the weight of the MWIS in the remaining graph remains unchanged. +(b) Suppose vertex v is a confined vertex, then either it is sheathed or the corre- +sponding confining set Sv ◀ AI. Moreover, if a vertex u ∈ Sv is also a confined +vertex with the corresponding confining set Su and v ∈ Su, then vertex set {u, v} +is a simultaneous set. +From Corollary 3.3, it can be known that the conflict analysis process in Defini- +tion 3.2 can be used to find the vertex that is sheathed or a simultaneous set. These +9 + +h +a +: +4 +3 +b +6 +11 +2 +g +d +9 +10 +m +c +5 +5 +k +5 +f +3 +8 +4 +eCITs will be used to design reduction rules in Section 4.1. In addition, by the prop- +erty of confined vertex, a fact is obvious: If confined vertex v such that {v} ◀ AI, +then the corresponding confining set Sv ◀ AI. We will exploit this state-preserving +result in the B&R algorithm to design a branching rule to search for a solution in +Section 4.2. +Next, we proceed to consider the assumption that a vertex is strongly sheathed. +In the MIS problem, the notion of mirror is given by means of such an assumption +and is very useful in practice [1]. We will generalize the notion of mirror to the +MWIS problem: For a vertex v ∈ V , a mirror of vertex v is a vertex u ∈ N 2(v) such +that w(v) ⩾ αw(G[N(v)\N(u)]). +Remark 3.4. When the weight of all vertices in the graph is 1, then α(G[N(v)\N(u)]) = +αw(G[N(v)\N(u)]) ⩽ w(v) = 1. This means that N(v)\N(u) induces a clique or is +an empty set, and this is exactly the definition that vertex u is the mirror of vertex +v in the MIS problem. +To make the concept of mirror more practical, we further generalize it to the case +of set, which leads to the following definitions: +Definition 3.5. Let set C be a vertex subset in the graph. If a vertex v ∈ C satisfies +the inequality: w(v) < αw(G[N(v)\C]), we call it a father of set C. Furthermore, +if there exists a vertex u ∈ N 2(v) such that w(v) ⩾ αw(G[N(v)\(C ∪ N(u))]), then +the father v is called an extending father of set C and vertex u is called a mirror of +vertex v. We use M(v) to denote the set of mirrors of vertex v. +By means of Definition 3.5, and under the assumption that a vertex is strongly +sheathed, the concept of ‘covered/uncovered vertices’ is given by the following conflict +analysis process: +Definition 3.6. Let v be a vertex in the graph. At the beginning, suppose set C := +{v} ◁ AC and repeating (i) until (ii) or (iii) are met: +(i) When set C has an extending father, extend set C by including the correspond- +ing set of mirrors to set C. +(ii) If there is a vertex u ∈ C such that w(u) ⩾ αw(G[N(u)\C]), in this case, the +upper bound lemma is not satisfied, then halt and vertex v is called an uncovered +vertex. +(iii) If set C has no extending father, then halt and return set Cv = C. In this case, +vertex v is called a covered vertex and vertex set Cv is called the covering set +of vertex v. +10 + +Figure 2: An example of uncovered vertex and a MWIS of this graph is {a, e, g, h, j, l}. +Starting with set C := {a}, from Definition 3.5, it can be seen that vertex a is an +extending father of set C and set {e, g, h} is the mirrors set of vertex a. Thus, set C +can be extended to: {a, e, g, h}. Then, vertex h is also an extending father of set C +and set {j, k, l} is the mirrors set of vertex h. So set C can be further expanded as: +{a, e, g, h, j, k, l}. At this time, we find that w(l) ⩾ αw(G[N(l)\C]), then halt and +conclude that vertex a is uncovered. +An example of uncovered vertex is given in Figure 2 and we find that vertex a +is an uncovered vertex. In addition, the properties of uncovered/covered vertices +are worth further study. From the vertex cover extension theorem, in the conflict +analysis process of Definition 3.6, for any extending father f of set C, ∀u ∈ M(f), +if set C ◁ AC, set C ∪ {u} ◁ AC always holds. Thus, under the assumption set +C := {v} ◁ AC, if vertex v is not an uncovered vertex, then a state-preserving +result can be obtained: The corresponding covering set Cv ◁ AC. Otherwise, the +upper bound lemma is not satisfied in the local structure of set C, which contradicts +hypothesis set C ◁ AC. So vertex v is inclusive. Also, assume that the two covered +vertices u, v and the corresponding covering set Cu, Cv satisfy: v ̸∈ N(u), u ∈ Cv and +v ∈ Cu. If vertex v is inclusive, we first remove N[v] from graph G. Since v ∈ Cu, +then vertex v is a mirror of an extending father of an intermediate state set C′ of +set Cu and the upper bound lemma cannot be satisfied in graph G[V \N[v]] at this +time. Thus, vertex u is an uncovered vertex of graph G[V \N[v]] and is inclusive in +this graph. So there exists a MWIS in graph G containing both vertex v and vertex +u. Moreover, if {v} ◁ AC, {u} ◁ AC is clearly satisfied. Thus, from the symmetry +of the relationship between vertex u and vertex v, it can be known that vertex set +{u, v} is a simultaneous set. These properties are summarized as follows. +Corollary 3.7. Let v be a vertex in the graph G. +(a) If vertex v is an uncovered vertex, then it is inclusive. After deleting N[v] from +the graph, the weight of the MWIS in the remaining graph satisfies: αw(G) = +11 + +h +d +b +a +J +6 +3 +5 +3 +4 +3 +g +c +k +e +2 +8 +6 +9 +4 +fαw(G[V \N[v]]) + w(v). +(b) If vertex v is a covered vertex. Then, either vertex v is inclusive or the cor- +responding covering set Cv ◁ AC. Also, if another covered vertex u with the +corresponding covering set Cu satisfies: v ̸∈ N(u), u ∈ Cv and v ∈ Cu, then +vertex set {u, v} is a simultaneous set. +Corollary 3.7 gives the following results: The conflict analysis process in Defi- +nition 3.6 can be applied to find the vertex that is inclusive or a simultaneous set. +In Section 4.1, we will use these CITs to design reduction rules. Besides, by the +property of covered vertex in (b) of Corollary 3.7, we can know a state-preserving +result: if the covered vertex v such that {v} ◁ AC, then the corresponding covering +set Cv ◁ AC. +3.2 +Reducible state-preserving technique +Similar to the first type of CIT, the reducible state-preserving technique utilizes the +assumption that a vertex is reducible, that is, assumes that a vertex is inclusive or +sheathed. With these assumptions, we can give state-preserving results similar to +the first type of CIT. Before that, we give the following definition. +Definition 3.8. Let sets IS and IC be two vertex subsets in the graph and set IS +is an independent set. +(a) A vertex u ∈ N(IS) is called an inferred child of set IS if it holds that +w(u) > w(IS ∩ N(u)). +Further, if there is only a unique independent set +IS∗ ⊆ N(u)\N[IS] that satisfies the inequality: w(u) ⩽ w(IS∩N(u))+w(IS∗), +we call the inferred child u an inferred extending child of set IS and vertex set +IS∗ is called an inferred satellite set of set IS. +(b) A vertex v ∈ IC is called an inferred father of set IC if it holds that w(v) ⩽ +αw(G[N(v)\IC]). An inferred father v is called an inferred extending father of +set IC if there exists a vertex u ∈ N 2(v) such that w(v) > αw(G[N(v)\(IC ∪ +N(u))]) and vertex u is called an inferred mirror of vertex v. Also, IM(v) is +used to denote its set of inferred mirrors. +By virtue of Definition 3.8 and the assumption that a vertex is inclusive or +sheathed, we can directly give the definitions of inferred confining set and inferred +covering set accordingly. +Definition 3.9. Suppose there are no unconfined vertex in the graph. Let v be a +vertex in the graph. Beginning with the assumption set IS := {v} ⊆ Iw. +12 + +(i) Only if set IS has an inferred extending child in N(IS), set IS can be extended +by including the corresponding inferred satellite set to set IS. +(ii) The above process halts if set IS has no inferred extending child in N(IS) and +return set ISv = IS. We call vertex set ISv is the inferred confining set of +vertex v. +Definition 3.10. We assume that there are no uncovered vertex in graph. Let v be +a vertex in the graph. Starting with the assumption set IC := {v} ⊆ V Cw. +(i) While set IC has an inferred extending father, extend set IC by including the +corresponding set of inferred mirrors to set IC. +(ii) The above process halts if set IC has no inferred extending father and return +set ICv = IC. We call vertex set ICv is the inferred covering set of vertex v. +Examples of inferred confining set and inferred covering set are given in Fig- +ure 3. +By the process in Definition 3.9, we can find the inferred confining set +ISa = {a, c, e, j, g, h, k} of vertex a. +Similarly, according to the process in Defi- +nition 3.10, we can find the inferred covering set ICd = {b, d, f, i, l} of vertex d. +Moreover, from the independent set extension theorem and the vertex cover exten- +sion theorem, we can directly obtain the following Corollary: +Corollary 3.11. Let v be a vertex in the graph. +(a) If {v} ⊆ Iw, then the corresponding inferred confining set ISv ⊆ Iw. +(b) Suppose {v} ⊆ V Cw, then the corresponding inferred covering set ICv ⊆ V Cw. +From (a) of Corollary 3.11, under the premise {v} ⊆ Iw, the state-preserving +result can be obtained: ISv ⊆ Iw. We will integrate this result into the local search +process of heuristic algorithm in Section 4.3. In addition, (b) of Corollary 3.11 also +gives a similar state-preserving result result: If {v} ⊆ V Cw, then the corresponding +inferred covering set ICv ⊆ V Cw. This result can be used to design a branching rule +to search for a solution in Section 4.2. +Furthermore, during the branching process of the B&R algorithm, it is assumed +that a vertex v is selected for branching. Inspired by the successful application of +packing constraints in the MIS problem, we extend them to the MWIS problem and +propose the concept of “weight packing constraint”. +When assuming that vertex v is inclusive, if ∃u ∈ N(v) such that w(u) ⩾ w(v), +let N +(u) = N(u)\N[v]. To avoid obtaining another MWIS by adding vertex u to +the independent set and removing vertices in N(u) from the independent set, by the +13 + +Figure 3: Examples of inferred confining set and inferred covering set. A MWIS for +this graph is {a, c, e, g, h, j, k}. We first search for the inferred confining set ISa of +vertex a. Let set IS := {a}, it can be seen from (a) of Definition 3.8 that vertex b +is an inferred extending child of set IS and set {c} is an inferred satellite set of set +IS. Thus, set IS can be extended to: {a, c}. Further, vertex d is also an inferred +extending child of set IS and set {e, j} is the corresponding inferred satellite set. So +set IS can be further extended to: {a, c, e, j}. At this time, it can be found that both +vertex f and vertex i are inferred extending children of set IS. Then, both vertex +set {g, h} and vertex set {k} are the corresponding inferred satellite sets. Finally, +the inferred confining set of vertex a can be found as: ISa = {a, c, e, j, g, h, k}. +Furthermore, we continue to search the inferred covering set ICd of vertex d. Let +set IC := {d}, according to (b) of Definition 3.8, vertex d is an extending father of +set IC and set {b, f, i} is its inferred mirrors set. Then, set IC can be extended as: +{b, d, f, i}. Next, it can be found that vertex b is an extending father of set IC and set +{l} is its inferred mirrors set. Thus, set IC can be further extended as: {b, d, f, i, l}. +Finally, the inferred covering set of vertex d can be found as: ICd = {b, d, f, i, l}. +. +upper bound lemma, the following state-preserving result needs to be guaranteed to +hold: +w(v) + +� +z∈N+(u) +w(z)(1 − xz) > w(u). +The 0-1 integer variable xz is used to indicate whether vertex z ∈ N +(u) is in the +independent set, and xz = 0 means it is in the independent set, otherwise it is not. +Thus, a weight packing constraint can be created as shown below: +� +z∈N+(u) +w(z)xz < +� +z∈N+(u) +w(z) − (w(u) − w(v)). +(3.1) +When assuming that vertex v is sheathed, to avoid that a MWIS containing it can +be found by modifying its state, by means of the upper bound lemma, the following +14 + +d +C +e +9 +b +9 +3 +3 +11 +4 +a +8 +10 +5 +3 +6 +3 +10 +h +！ +kstate-preserving result needs to be satisfied: +� +z∈N(v) +w(z)(1 − xz) > w(v). +So a weight packing constraint can also be created as follows: +� +z∈N(v) +w(z)xz < +� +z∈N(v) +w(z) − w(v). +(3.2) +These constraints will be kept and managed while the algorithm is searching for +a solution, and we only need to search all branches satisfying these constraints, since +no better solution exists in the remaining branches, thus narrowing the search space. +Let � +z∈S +w(z)xz < k be a weight packing constraint such that set S is non-empty. +When a vertex z is found to be inclusive, for each constraint that includes variable +xz, we delete the variable on the left side of the constraint and keep the right side +of the constraint unchanged. When a vertex z is inferred to be sheathed, for each +constraint that contains variable xz, we delete the variable on the left side of the +constraint and decrease the weight of vertex z on the right side of the constraint. +In the process of keeping and managing these constraints, some properties of causal +inference are mined, which can be divided into the following three cases. +(a) When there is a constraint whose right-hand term k is less than or equal to 0, +then we can directly prune subsequent searches from the current branch vertex. +(b) When there is a constraint whose right-hand term k is less than or equal to +the weight of any vertex in set S, if this set is not an independent set, we can +prune subsequent searches from the current branch vertex. If not, the vertices +in set S will be included in the independent set. +In addition, some new weight packing constraints can also be introduced. Sup- +pose there is a vertex p ∈ N(S) such that w(p) ⩾ w(N(p) ∩ S), let N +(p) = +N(p)\N[S], by the upper bound lemma, the following state-preserving result +needs to be guaranteed: +w(N(u) ∩ S) + +� +z∈N+(u) +w(z)(1 − xz) > w(u). +Therefore, we can introduce the following weight packing constraint: +� +z∈N+(p) +w(z)xz < +� +z∈N+(p) +w(z) − (w(p) − w(N(p) ∩ S)). +(3.3) +15 + +(c) When there is a constraint whose right-hand term k > 0 and there is vertex +u ∈ N(S) such that +� +z∈N(u)∩S +w(z) ⩾ k, it can be inferred that vertex u is +sheathed to ensure that this constraint holds. In addition, in order to ensure +that the current state-preserving result is valid, similar to constraint (3.2), the +following constraint needs to be introduced: +� +z∈N(u) +w(z)xz < +� +z∈N(u) +w(z) − w(u). +(3.4) +The above properties of causal inference provide new pruning search techniques for +the B&R algorithm and can simplify the graph. We will integrate these techniques +into B&R algorithm in Section 4.2. +4 +Integrate CITs into Existing Algorithmic Frame- +works +We next describe in detail how CITs in Section 3 are integrated into the existing +algorithmic frameworks. Section 4.1 introduces how to apply the first type of CIT +to the reduction algorithm. Further, integrating the resulting reduction algorithm +and the state-preserving results of two types of CITs into B&R algorithm will be +presented in Section 4.2, and Section 4.3 will introduce the application of the state- +preserving results of the second type of CIT to the local search process of heuristic +algorithm. +4.1 +The Causal Reduce +We first introduce how to design reduction rules with the first type of CIT and +how to integrate them into the existing reduction algorithm. From the property of +unconfined vertex in Corollary 3.3 and the property of uncovered vertex in Corol- +lary 3.7, the following reduction rules that can directly determine whether a vertex +is reducible are given first: +• Rule I: Check whether a vertex v is unconfined or confined by the procedure +in Definition 3.2, and if it is unconfined, remove vertex v directly from the +graph. +• Rule II: Use the procedure in Definition 3.6 to check whether a vertex v is cov- +ered or uncovered, and if it is uncovered, include vertex v into the independent +set and remove N[v] from the graph. +16 + +Before further introducing how to utilize the first type of CIT to design reduction +rules, we first give an important property about simultaneous set mentioned in [27]: +A simultaneous set S can be contracted by removing all vertices in set S from the +graph and introducing a vertex v∗ such that it is adjacent to all vertices in N(S) +with weight w(v∗) = w(S), while the weight of the MWIS in the remaining graph +remain unchanged. +Next, we will design reduction rules on simultaneous set through the first type of +CIT, and give the following definitions by the results of the simultaneous set given +in (b) of Corollary 3.3 and (b) of Corollary 3.7. +Definition 4.1. Let u, v be two vertices in the graph. +(a) Suppose vertices u and v be two confined vertices with confining set Su and Sv. +If u ∈ Sv and v ∈ Su, then set {u, v} is called a confining simultaneous set. +(b) Assume that vertices u and v be two covered vertices with covering set Cu and +Cv. Set {u, v} is called a covering simultaneous set if u ∈ Cv and v ∈ Cu. +From Definition 4.1, we have the following rules: +• Rule III: If there are two confined vertices that constitute a confining simul- +taneous set, then merge them. +• Rule IV: Merge two covered vertices u and v if they form a covering simulta- +neous set. +Next, we will describe how to integrate our reduction rules into an existing re- +duction algorithm—Reduce proposed by [27]. Reduce consists of seven steps. The +reduction rules used in these steps exploit the sufficient conditions that a vertex is +reducible. It executes these steps incrementally, which means that the next step is +only executed when all previous steps are no longer applicable. Thus, if the graph +is changed, it will go back to the first step. Notably, our reduction rules I and III +are further generalization of the reduction rules used in step 5 of Reduce. So, we +can combine our reduction rules I and III into one step to replace step 5 in Reduce +and label this step as Remove Unconfined & Contract Confining. Similarly, +we can also integrate our reduction rules II and IV into another new step in the +reduction algorithm, called Remove Uncovered & Contract Covering. +• Remove Unconfined & Contract Confining: Check whether a vertex is +unconfined or confined. If it is confined, apply Rule I to remove it; If not, use +Rule III to contract the corresponding confining simultaneous set when it can +be found. +17 + +• Remove Uncovered & Contract Covering: If a vertex is checked to be +uncovered, use Rule II to reduce it. Otherwise, if the corresponding covering +simultaneous set can be found, use Rule IV to merge it. +Figure 4: Casual Reduce: Given an input graph G, each step of the algorithm +is executed sequentially and the graph changes, immediately go back to the first +step. When all steps are completed and the graph no longer changes, return to the +remaining graph kernel. +Thus, a new reduction algorithm called Causal Reduce can be obtained by using +Remove Unconfined & Contract Confining to replace step 5 of Reduce and +adding Remove Uncovered & Contract Covering between Remove Uncon- +fined & Contract Confining and step 6 of Reduce, which is shown in Figure 4. +We will use Causal Reduce(G) = (K, c) to represent the processing of this algo- +rithm on a given input graph G. The processing result of this algorithm consists of +two parts: One is the remaining graph called kernel K and the other is the weight of +the vertex set contained in the MWIS obtained by inference. It’s worth noting that +the reduction algorithm Causal Reduce may not resolve all instances directly, but +it can be used as a preprocessing for heuristic and exact algorithm. +4.2 +The Causal B&R Solver +Before introducing how to integrate our CITs into B&R algorithm, we briefly intro- +duce the state-of-the-art exact algorithm Solve proposed by [27]. Solve is based on +the idea of B&R algorithm, which first apply reduction algorithm Reduce to reduce +the instance. Then, apply branching rule by virtue of the property of the confining +set and perform reduction algorithm Reduce in every branch of the search tree to +18 + +Step 4 of Reduce +Remove Unconfined & Contract Confining +Step 3 of Reduce +Remove Uncovered& Contract Covering +Graph Change +Step 2 of Reduce +Step 6 of Reduce +Step 1 of Reduce +Step 7 of Reduce +G = (V, E, w) +Kernel Kfind a solution. During the searching, it uses a standard technique based on finding +upper and lower bounds to prune the search tree and take the best solution weight +Wb currently found in the algorithm as the lower bound. Initially, let Wb be the +weight of the solution obtained by heuristic algorithm on the kernel K, and update +Wb once a better solution is obtained in the algorithm. The heuristic algorithm, +denoted by Greedy(G), is a greedy algorithm that iteratively selects a vertex in +order of some measure and removes its closed neighbor set from the graph. In each +searching branch, it uses a heuristic method to find an upper bound Wub of the op- +timal solution weight of the current graph, which is based on weight clique covers +and is denoted by UpperBound(G). If the current best solution weight Wb is not +smaller than Wub, then there is no better solution in this searching branch and it can +be discarded directly. +Algorithm 1 The Causal B&R Solver(G) +Require: A vertex weight graph G = (V, E, w); +Ensure: The weight of a MWIS of G. +1: Initialization of global variable Wb: Wb ← 0; +2: if weight packing constraints have been created then +3: +while True do +4: +(K, c) ← Causal Reduce(G); +5: +check constraints(); +6: +if existence constraints are not satisfied then +7: +return Wb; +8: +else if graph is simplified then +9: +continue; +10: +else +11: +break; +12: else +13: +(K, c) ← Causal Reduce(G); +14: Wb ← max{Wb, c + Greedy(K)}; +15: if c + UpperBound(G) ⩽ Wb then +16: +return Wb; +17: Pick up a vertex v of maximum degree and compute the confining set Sv and the inferred covering set ICv; +18: create weight packing constraint (3.1) and Wb ← max{Wb, c + w(Sv) + Causal B&R Solver(K − N[Sv])}; +19: create weight packing constraint (3.2) and Wb ← max{Wb, c + Causal B&R Solver(K − ICv)}; +20: return Wb; +Our CITs will be integrated into two parts of Solve, resulting in a new exact al- +gorithm called Causal B&R Solver. The first part is that we will use our reduction +algorithm Causal Reduce to reduce the instance to get the kernel K, and perform +the reduction algorithm on each branch of the search tree. The second part is that +we will make use of the state-preserving results of two types of CITs during the +branching process. Similar to the idea of Solve in [27], using property of confining +set to the branching process, when choosing a vertex with the maximum degree to +branch, the state-preserving results of (b) of Corollary 3.11 and (b) of Corollary 3.3 +19 + +will be used in this part. This means that during branching, we either remove the +inferred covering set of the branching vertex from the graph or include the confining +set of the branching vertex into the independent set. Furthermore, we will create +weight packing constraint (3.2) while removing the inferred covering set of branching +vertex. Similarly, we will also create weight packing constraint (3.1) when includ- +ing the confining set of branching vertex into the independent set. We will keep and +manage these weight packing constraints when searching for solutions in each branch +of the search tree. Specifically, another step called check constraints is added after +the last step of Casual Reduce. In this step, for each weight packing constraint, we +will check whether the constraint holds and whether the graph can be simplified by +the causal inference properties of that constraint. If any constraint is violated, the +searching branch will be skipped. If the graph can be simplified, Causal Reduce +will continue to execute after reducing the graph. If none of the above conditions are +met, the subsequent process will be performed. The main steps of Causal B&R +Solver are listed in Algorithm 1. +4.3 +The Causal Search +After taking our reduction algorithm Causal Reduce as a preprocessing, we ap- +ply the state-preserving result of second type of CIT to the local search process +of heuristic algorithm DynWVC2 [6] to solve the complementary problem of the +MWIS problem—the MWVC problem, which leads to a new algorithm called Causal +Search. +Algorithm 2 The basic framework of DynWVC2 algorithm. +Require: A vertex weight graph G = (V, E, w), the cutoff time of the running T; +Ensure: A vertex cover of G. +1: V C ← Construct(); +2: V C∗ ← V C; +3: while elapsed time 0 there is an ε-chain of length m connecting x to itself for some m (m depends +on ε > 0). The chain recurrent set, denoted by Rf, is the set of all chain recurrent +points of f. Define the equivalence on Rf by the rule: x ∼ y if for every ε > 0 +there is are ε-chains connecting x to y and y to x. This equivalence relation defines +equivalence classes called chain components. +If the chain recurrent set of a diffeomorphism f is finite then it consists of +periodic points. A periodic point p ∈ Rf of period mp is said to be hyperbolic if +absolute values of all the eigenvalues of the Jacobian matrix +� +∂f mp +∂x +� +|p are not +equal to 1. If absolute values of all these eigenvalues are greater (less) than 1 then +p is called a sink (a source). Sinks and sources are called knots. If a periodic point +is not a knot then it is called a saddle. +Let p be a hyperbolic periodic point of a diffeomorphism f whose chain recur- +rent set is finite. The Morse index of p, denoted by λp, is the number of eigen- +values of Jacobian matrix whose absolute values are greater than 1. The stable +manifold W s +p = {x ∈ M n : +lim +k→+∞ d(f kmp(x), p) = 0} and the unstable manifold +W u +p = {x ∈ M n : +lim +k→+∞ d(f −kmp(x), p) = 0} of p are smooth manifolds diffeo- +morphic to Rλp and Rn−λp, respectively. Stable and unstable manifolds are called +invariant manifolds. A connected component of the set W u +p \ p (W s +p \ p) is called +a unstable (stable) separatrice of p. +A diffeomorphism f : M n → M n is called a Morse-Smale diffeomorphism if +1. its chain recurrent set Rf consists of finite number of hyperbolic points; +2. for any two points p, q ∈ Rf the manifolds W s +p , W u +q intersect transversally. + +Quasi-energy function for Pixton diffeomorphisms +3 +C Conley in [3] gave the following definition: a Lyapunov function for a Morse- +Smale diffeomorphism f : M n → M n is a continuous function ϕ : M n → R +satisfying +– ϕ(f(x)) < ϕ(x) if x /∈ Rf; +– ϕ(f(x)) = ϕ(x) if x ∈ Rf. +Notice that every Morse-Smale diffeomorphism f has a Morse-Lyapunov func- +tion 1, i.e. a Lyapunov function ϕ : M n → R which is a Morse function such that +each periodic point p ∈ Rf is its non-degenerate critical point of index λp with +Morse coordinates (Vp, φp : y ∈ Vp �→ (x1(y), . . . , xn(y)) ∈ Rn and +φ−1 +p (Ox1 . . . xλp) ⊂ W u +p , φ−1 +p (Oxλp+1 . . . xn) ⊂ W s +p . +(∗) +If the function ϕ has no critical points outside Rf then following [15] we call it +the energy function for the Morse-Smale diffeomorphism f. +The proof of existence of an energy Morse function for a Morse-Smale diffeo- +morphism of the circle is an easy exercise. D. Pixton [15] in 1977 proved that +every Morse-Smale diffeomorphism of a surface has an energy function. There he +also constructed an example of a Morse-Smale diffeomorphism on the 3-sphere +which admits no energy function. The obstacle to existence of an energy function +in his example was the wild embedding of the saddle separatrices in the ambient +manifold (i.e. the closure of the separatrice is not a submanifold of the ambient +space). From [11] it follows that there are Morse-Smale diffeomorphisms with no +energy function on manifolds of any dimension n > 2. Therefore, following [7] +for a Morse-Smale diffeomorphism f we call a Morse-Lyapunov function with the +minimal number of critical points (denote it by ρf ) a quasi-energy function. Notice +that ρf is a topological invariant, i.e. if two diffeomorphisms f, f ′ : M n → M n +are topologically conjugate (that is there is a diffeomorphism h : M n → M n such +that h ◦ f = f ′ ◦ h) then ρf = ρf′ . +In this paper we give a lower estimate of ρf for Pixton diffeomorphisms. The +class of Pixton diffeomorphisms P is defined in the following way. Every diffeo- +morphism f ∈ P is a Morse-Smale 3-diffeomorphism whose chain recurrent set +consists of four points: one source, one saddle and two sinks (for details see section +2). Notice that Pixton’s example is a diffeomorphism of this class. According to [2] +the class of topological conjugacy of a diffeomorphism f ∈ P is completely defined +by the equivalence class of the Hopf knot Lf, i.e. the knot in the generating class +of the fundamental group of the manifold S2 × S1 (see Proposition 1). Moreover, +any Hopf knot can be realized as a Pixton diffeomorphism. +Recall that a knot in S2 × S1 is a smooth embedding γ : S1 → S2 × S1 or the +image of this embedding L = γ(S1). Two knots L, L′ are said to be equivalent if +there is a homeomorphism h : S2 × S1 → S2 × S1 such that h(L) = L′. Two knots +1 This function can be constructed, for example, by suspension. Consider the topological +flow ˆft on the manifold Mn × R defined by ˆft(x) = x + t. Define the diffeomorphism g : +Mn × R → Mn × R by g(x, τ) = (f(x), τ − 1) and let G = {gk , k ∈ Z} and W = (Mn × R)/G. +Denote by pW : Mn × R → W the natural projection and denote by ft the flow on W defined +by ft(x) = pW ( ˆft(p−1 +W (x))). The flow ft is called the suspension over f. By construction the +chain recurrent set of ft consists of the finite number of periodic orbits δi = pW (Oi × R), i ∈ +{1, . . . , kf} and this means that the suspension ft is a Morse-Smale flow. A Lyapunov function +for these flows is constructed in [12]. Then the restriction of this function on M is the desired +Lyapunov function for f. + +4 +Timur Medvedev, Olga Pochinka +γ, γ′ are smoothly homotopic if there exists a smooth map Γ : S1 ×[0, 1] → S2 ×S1 +such that Γ(s, 0) = γ(s) and Γ(s, 1) = γ′(s) for every s ∈ S1. If Γ|S1×{t} is an +embedding for every t ∈ [0, 1] then the knots are said to be isotopic. +Any Hopf knot L ⊂ S2 × S1 is smoothly homotopic to the standard Hopf +knot L0 = {s} × S1 (see, for example, [9]) but generally it is neither isotopic +nor equivalent to it. B. Mazur [10] constructed the Hopf knot LM which we call +the Mazur knot and which is non-equivalent and non-isotopic to L0 (see Fig. 2). +It follows from the results of [1] that there exists a countable family of pairwise +Fig. 2 Two non-isotopic and non equivalent Hopf knots L0 and LM: a) the standard Hopf +knot L0; b) the Mazur knot LM +non-equivalent Hopf knots Ln, n ∈ N which are generalized Mazur knots (Fig. 3). +Fig. 3 A generalized Mazur knot Ln +According to [6] a Pixton diffeomorphism f admits an energy Morse function +if and only if the knot Lf is trivial (i.e. equivalent to the standard one). If the + +Quasi-energy function for Pixton diffeomorphisms +5 +knot Lf is not trivial then the number ρf of the critical points of a quasi-energy +Morse function of f is evidently even and +ρf ⩾ 6. +The main result of this paper is the proof of Theorem 1. +Theorem 1 Let f be a Pixton diffeomorphism (f ∈ P) and let Ln, n ∈ N be +its knot. Then the number ρf of critical points of a quasi-energy function of f is +calculated by2 +ρf = 4 + 2n. +2 Construction of Pixton diffeomorphisms +In dynamics a wild Artin-Fox arc was for the first time introduced by D. Pixton +in [15] where he constructed a Morse-Smale diffeomorphism on the 3-sphere with +the unique saddle whose invariant manifolds form an Artin-Fox arc. We give the +modern construction of these diffeomorphisms following Ch. Bonatti and V. Grines +[2] where Pixton diffeomorphisms were also classified (see also [8], [11]). +For x = (x1, . . . , xn) ∈ Rn denote ||x|| = +� +x2 +1 + · · · + x2n. Let h : R3 → R3 +be the diffeomorphism defined by h(x1, x2, x3) = +� x1 +2 , x2 +2 , x3 +2 +� +. Define the map +p : R3 \ O → S2 × S1 by +p(x1, x2, x3) = +� x1 +||x||, x2 +||x||, log2(||x||) +(mod 1) +� +. +Let L ⊂ (S2 ×S1) be a Hopf knot and let U(L) be its tubular neighborhood. Then +the set ¯L = p−1(L) is the h-invariant arc in R3 and U(¯L) = p−1(U(L)) is its +h-invariant neighborhood diffeomorphic to D2 × R1 (Fig. 4). +Let C = {(x1, x2, x3) ∈ R3 +: x2 +2 + x2 +3 ⩽ 4} and let gt : C → C be the flow +defined by +gt(x1, x2, x3) = (x1 + t, x2, x3). +Then there is a diffeomorphism ζ : U(L) → C that conjugates h|U(L) and g = g1|C. +Define the flow φt on C by: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +˙x1 = +� +1 − 1 +9(x2 +1 + x2 +2 + x2 +3 − 4)2, +x2 +1 + x2 +2 + x2 +3 ⩽ 4 +1, +x2 +1 + x2 +2 + x2 +3 > 4 +˙x2 = +� +� +� +� +� +x2 +2 +� +sin +� π +2 +� +x2 +1 + x2 +2 + x2 +3 − 3 +�� +− 1 +� +, +2 < x2 +1 + x2 +2 + x2 +3 ⩽ 4 +−x2, +x2 +1 + x2 +2 + x2 +3 ⩽ 2 +0, +x2 +1 + x2 +2 + x2 +3 > 4 +˙x3 = +� +� +� +� +� +x3 +2 +� +sin +� π +2 +� +x2 +1 + x2 +2 + x2 +3 − 3 +�� +− 1 +� +, +2 < x2 +1 + x2 +2 + x2 +3 ⩽ 4 +−x3, +x2 +1 + x2 +2 + x2 +3 ⩽ 2 +0, +x2 +1 + x2 +2 + x2 +3 > 4. +By construction the diffeomorphism φ = φ1 has two fixed points: the saddle +P(1, 0, 0) and the sink Q(−1, 0, 0) (Fig. 5), both being hyperbolic. One unstable +2 For n = 1 Theorem 1 is proved in [7]. + +6 +Timur Medvedev, Olga Pochinka +L +L +_ +U(L) +_ +Fig. 4 Suspension of a Hopf knot +1 +-1 +2 +O +Fig. 5 Trajectories of the flow φt +separatrice of the saddle P coincides with the open interval +� +(x1, x2, x3) ∈ R3 : |x1| < 1, x2 = x3 = 0 +� +in the basin of the sink Q while the other coincides with the ray +� +(x1, x2, x3) ∈ R3 : x1 > 1, x2 = x3 = 0 +� +. +Notice that φ coincides with the diffeomorphism g = g1 outside the ball {(x1, x2, x3) ∈ +C : x2 +1 + x2 +2 + x2 +3 ⩽ 4}. +Define the diffeomorphism ¯fL : R3 → R3 so that ¯fL coincides with h outside +U(L) and it coincides with ζ−1φζ on U(L). Then ¯fL has in U(L) two fixed points: +the sink ζ−1(Q) and the saddle ζ−1(P), both being hyperbolic. The unstable +separatrice of the saddle ζ−1(P) lies in L (Fig. 6). +Now project the dynamics onto the 3-sphere. Denote by N(0, 0, 0, 1) the North +Pole of the sphere S3 = {x = (x1, x2, x3, x4) : ||x|| = 1}. For every point x ∈ (S3 \ +{N}) there is the unique line passing through N and x in R4. This line intersects + +Quasi-energy function for Pixton diffeomorphisms +7 +R +Fig. 6 The phase portrait of the diffeomorphism ¯fL +R3 in exactly one point ϑ+(x) (Fig. 7). The point ϑ+(x) is the stereographic +projection of the point x. One can easily check that +ϑ+(x1, x2, x3, x4) = +� +x1 +1 − x4 , +x2 +1 − x4 , +x3 +1 − x4 +� +. +Thus, the stereographic projection ϑ+ : S3 \ {N} → R3 is a diffeomorphism. +N +x +��(x) +Fig. 7 The stereographic projection. +By construction ¯fL coincides with h in some neighborhood of the point O and +in some neighborhood of the infinity. Therefore, it induces on S3 the Morse-Smale +fL(x) = +� +ϑ−1 ++ ( ¯fL(ϑ+(x))), x ̸= N; +N, x = N +. +It follows directly from the construction that the non-wandering set of fL consists +of exactly four fixed hyperbolic points: two sinks ω = ϑ−1 ++ (ζ−1(Q)), S, one saddle +σ = ϑ−1 ++ (ζ−1(P)) and one source N. We say the constructed diffeomorphism to +be model and it is of Pixton class. + +8 +Timur Medvedev, Olga Pochinka +Proposition 1 ([2]) +– Any diffeomorphism f ∈ P is topologically conjugate to some model diffeomor- +phism fL. +– Two model diffeomorphisms fL, fL′ are topologically conjugate if and only if +their knots L, L′ are equivalent. +3 Genus of Hopf knot +In this section we introduce the notion of genus for a Hopf knot and use it to +estimate the number of critical points of the quasi-energy function of the Pixton +diffeomorphism defined by this knot. +Let L be a Hopf knot and let ¯L = p−1(L) be its cover in R3\O. We say a closed +orientable surface Σ ⊂ S2 × S1 to be a secant surface of the knot L if it intersects +L in a unique point and there is an h-compressible 3-manifold QΣ ⊂ R3 (that is +h(QΣ) ⊂ int QΣ) with the boundary ¯Σ such that Σ = p( ¯Σ) and the intersection +¯L∩ ¯Σ is the unique point ¯y. The minimally possible genus gL of the secant surface +is called the genus of the knot L. The secant surface of L of genus gL is said to be +minimal. +Lemma 1 If Σ is a minimal secant surface of the knot L then the surface ¯Σ \ ¯y +is non-compressible in R3 \ (O ∪ ¯L), i.e. any simple closed curve c ⊂ int ( ¯Σ \ ¯y) is +contractible on ¯Σ\ ¯y if it bounds a smoothly embedded 2-disk D ⊂ int (R3\(O∪ ¯L)) +such that D ∩ ( ¯Σ \ ¯y) = ∂D = c. +Proof Let Σ be a minimal secant surface of L and let ¯y be the unique point of +the intersection ¯L ∩ ¯Σ. Assume the opposite: the surface ¯Σ \ ¯y is compressible in +R3 \ (O ∪ ¯L). Then there is a non-contractible simple closed curve c ⊂ int ( ¯Σ \ ¯y) +and there is the smoothly embedded 2-disk D ⊂ int (R3 \ (O ∪ ¯L)) such that +D ∩ ( ¯Σ \ ¯y) = ∂D = c (see, for example, [14]). Then we have two possibilities: +(int D) ∩ +� � +k∈Z +hk( ¯Σ) +� += ∅, +(1) +(int D) ∩ +� � +k∈Z +hk( ¯Σ) +� +̸= ∅. +(2) +In case (1) two subcases are possible: (1a) D ⊂ QΣ, (1b) D ⊂ (R3 \ int QΣ). For +case 1a) let N(D) ⊂ QΣ be a tubular neighborhood of the disk D. Then exactly +one connected component of the set QΣ \ int N(D) intersects ¯L. According to (1) +this component is h-compressible and its boundary intersects ¯L at a unique point. +The projection of this boundary into S2 × S1 is, therefore, the secant surface of +L of genus less than gL. This contradicts the fact that the surface Σ is minimal. +In case 1b) let N(D) ⊂ (R3 \ int QΣ) be a tubular neighborhood of D. Then due +to (1) the set QΣ ∪ N(D) is h-compressible and its boundary intersects ¯L at a +unique point. The projection of this boundary into S2 ×S1 is, therefore, the secant +surface of L of genus less than gL and we have the same contradiction. + +Quasi-energy function for Pixton diffeomorphisms +9 +In case (2) without loss of generality assume the intersection int D∩( � +k∈Z +hk( ¯Σ)) +to be transversal and denote it by Γ. Let γ be a curve from Γ. We say the curve +γ to be innermost if it is the boundary of the disk Dγ ⊂ D such that int Dγ +contains no curves of Γ. Consider this innermost curve γ ⊂ f k(Σ). There are +two subcases: a) γ is essential on f k(Σ) and b) γ is contractible on f k(Σ). In +case a) the arguments of the case (1) apply for the body f k(QΣ) and the disk +Dγ and we get the contradiction to the minimality of the surface Σ. In case b) +denote by dγ ⊂ f k(Σ) the 2-disk bounded by γ and denote by Bγ ⊂ (R3 \ O) +the 3-ball bounded by the 2-sphere Dγ ∪ dγ. Consider: b1) Bγ ⊂ f k(QΣ) and b2) +Bγ ⊂ (R3 \ int f k(QΣ)). For b1) let N(Bγ) ⊂ f k(QΣ) be a tubular neighborhood +of Bγ. Then the set QΣ \ int N(Bγ) is h-compressible because the curve γ lies in +its interior and the boundary of QΣ \ int N(Bγ) intersects ¯L at a unique point. +The projection of this boundary into S2 ×S1 is, therefore, the secant surface of the +knot L of genus gL for which the number of connected components of the set Γ is +less. We get the same result for b2) for the set QΣ ∪ N(Bγ). Thus, iterating the +process we come either to the case a) or to the case (1) and get a contradiction. +Lemma 2 For any diffeomorphism f ∈ P the following estimation holds +ρf ⩾ 4 + 2gLf . +(3) +Proof Since Proposition 1 is true and since the number ρf of the critical points of a +quasi-energy function of f ∈ P is invariant, from now on we consider model Pixton +diffeomorphismsfL with the Hopf knot L. Denote by ℓ the non-stable separatrice +of the saddle σ lying in the basin of the sink S. Let +pS : W s +S \ S → S2 × S1 +be the natural projection sending a point w ∈ (W s +S\S) to the point p(f kw(w)), f kw(w) ∈ +VS. Since the diffeomorphism fL coincides with the homothety h in some neigh- +borhood VS of S, the natural projection pS is well defined and pS(ℓ) = L by +construction. +Consider an arbitrary Morse-Lyapunov function ϕ : S3 → R of the diffeomor- +phism fL. To be definite let ϕ(S) = 0, ϕ(σ) = 1 and ϕ(N) = 3. From the definition +of the Morse-Lyapunov function it follows that ϕ|ℓ monotonically decreases in some +neighborhood of the saddle σ. Therefore, there is ε1 ∈ (0, 1) such that the interval +(1 − ε1, 1) contains no critical values of ϕ and the connected component ¯Σ1 of the +level set ϕ−1(1 − ε1) intersects the separatrice ℓ at the unique point. Denote this +point by w1. +Let ¯Q1 be the connected component of the set ϕ−1([0, 1 − ε1]) which contains +the segment [w1, S] of the closure of the separatrice ℓ. Since ϕ decreases along +the trajectories of f, the values of ϕ on W s +σ are greater than 1. Therefore, the +manifold ¯Q1 lies in the manifold W s +S diffeomorphic to R3. Let the function ϕ| ¯ +Q1 +have kq, q ∈ {0, . . . , 3} critical points of index q. Due to [5, Theorem 6.1] on the +manifold ¯Q1 there exists a self-indexing Morse function ψ (the value of the function +in a critical point equals the index of this point) which has kq critical points of +index q and which is constant on ∂ ¯Q1. Thus, the manifold ¯Q1 is the surface ˜Q1 of + +10 +Timur Medvedev, Olga Pochinka +genus g1 = 1 + k1 − k0 with attached handles of indexes 2 and 3. Then the genus +of any surface of the set ∂ ¯Q1 cannot be greater than g1. +On the other hand, the number of critical points of ϕ| ¯ +Q1 is not less than k0+k1. +If k0 ⩾ 1 and g1 = 1 + k1 − k0 then one gets k0 + k1 = g1 + 2k0 − 1 ⩾ g1 + 1. Thus, +ϕ| ¯ +Q1 has at least g1 + 1 critical points. +Denote by ¯Σ1 the connected component of ∂ ¯Q1 which intersects the separatrice +ℓ. Then the surface ¯Σ1 divides the manifold W s +S ∼= R3 into two parts, one of which +Q1 being an h-compressible body. This means that Σ1 = pS( ¯Σ1) is the secant +surface of L and, therefore, +g1 ⩾ gL. +Analogously, there is ε2 ∈ (0, 1) for which the interval (1, 1 + ε2) contains no +critical points of ϕ and the connected component ¯Q2 of the level set ϕ−1([0, 1+ε2)] +contains cl(W u +σ ) in its interior while the intersection ¯Q2 with W s +σ is the unique +2-disk. Due to construction the function ϕ| ¯ +Q2 has at least g1 + 3 critical points +and genus of the connected components of ∂ ¯Q2 is less or equals g1. Denote by ¯Σ2 +the connected component of ∂ ¯Q2 which intersects W s +σ and denote by g2 its genus. +The surface ¯Σ2 divides the manifold W u +N ∼= R3 into two parts, one of which Q2 +being an h−1-compressible body. Arguing as above one comes to conclusion that +the number of critical points of ϕ|Q2 is at least g2 +1. Therefore, the total number +of critical points of ϕ is greater or equal to +g1 + 3 + g2 + 1 ⩾ 4 + 2g1 ⩾ 4 + 2gLf . +4 The generalized Mazur knot Ln +In this section we show that the genus gLn of a generalized Mazur knot equals n. +At first we give a detailed description of construction of Ln. +4.1 Construction of the generalized Mazur knot Ln +Recall that h : R3 → R3 is the homothety defined by +h(x1, x2, x3) = +�x1 +2 , x2 +2 , x3 +2 +� +and p : R3 \ O → S2 × S1 is the natural projection defined by +p(x1, x2, x3) = +� x1 +||x||, x2 +||x||, log2(||x||) +(mod 1) +� +. +Consider the annulus +K = +� +(x1, x2, x3) ∈ R3 : 1 +4 ≤ x2 +1 + x2 +2 + x2 +3 ≤ 1 +� +bounded by the spheres +S2 = +� +(x1, x2, x3) ∈ R3 : x2 +1 + x2 +2 + x2 +3 = 1 +� +, h(S2). + +Quasi-energy function for Pixton diffeomorphisms +11 +Pick on the circle +S1 = +� +(x1, x2, x3) ∈ R3 : x2 +1 + x2 +2 = 1, x3 = 0 +� +pairwise distinct points α1, . . . , α2n+1 numbered in counter-clockwise order (Fig. 8). +Let ai, i ∈ {1, . . . , 2n} be the arc of the circle S1 bounded by αi, αi+1 whose inte- +rior contains no points of {α1, . . . , α2n+1}. Let B, Ai ⊂ int K, i ∈ {1, . . . , 2n} be +pairwise disjoint smooth arcs such that: +1. the boundary points of B are α2n+1, h(α1); the boundary points of A2j−1 +are α2j−1, α2j and the boundary points of A2j are h(α2j), h(α2j+1) for j ∈ +{1, . . . , n}; +2. the closed curves c2j−1 = cl(a2j−1 ∪ A2j−1), c2j = cl(h(a2j) ∪ A2j) bound the +2-disks d2j−1, d2j, the transversal intersection of these disks being the arc lj +with the boundary points b2j−1 = d2j−1 ∩ A2j and b2j = d2j ∩ A2j−1; +3. the arc cl(h(A1) ∪ A2 ∪ · · · ∪ h(A2n−1) ∪ A2n ∪ B) is smooth. +Fig. 8 Construction of the knot Ln +Let +¯Ln = +� +k∈Z +hk(B ∪ A1 ∪ · · · ∪ A2n), Ln = p(¯Ln). +4.2 The genus of the knot Ln +Lemma 3 The genus gLn of the knot Ln equals n. + +12 +Timur Medvedev, Olga Pochinka +Fig. 9 A secant surface of Ln of genus n +Proof Since there is a secant surface of Ln of genus n, we have gLn ⩽ n (Fig. 9). +Now we show that gLn ⩾ n. To that end we prove that for Ln there exists a +minimal secant surface Σ such that ¯Σ ⊂ K and ¯Ln ∩ ¯Σ = h(α1). +Indeed, let Σ0 be some minimal secant surface of Ln. Then there exists the +connected component ¯Σ0 of p−1(Σ0) such that it intersects the curve ¯Ln at the +point ¯y0 situated on ¯Ln between α1, h(α1) and that bounds the h-compressible +body QΣ0. Without loss of generality let ¯y0 = h(α1) (otherwise the desired surface +is constructed by removing the tubular neighborhood of the arc [¯y0, h(α1)] ⊂ ¯Ln +from QΣ0). +Denote by k+, k− ⩾ 0 the maximal integers for whichf k( ¯Σ0)∩ ¯Σ0 ̸= ∅, f −k( ¯Σ0)∩ +¯Σ0 ̸= ∅, k ⩾ 0, respectively. If k+ = k− = 0 then ¯Σ0 is the desired surface. Other- +wise we show the way to decrease by 1 the number k+ > 0 (for k− the arguments +are the same) using isotopy of the secant surface. +Notice that ¯Σ0 ∩ f k+(c2j−1) = ∅, j ∈ {1, . . . , n}. Without loss of generality let +the intersection Γ = +n� +j=1 +f k+(d2j−1) ∩ ¯Σ0 be transversal. Let γ be a curve from +Γ. Then γ bounds the unique disk Dγ ⊂ f k+(d2j−1). There are two possibilities: +1) b2j−1 /∈ Dγ, 2) b2j−1 ∈ Dγ. In case 1) we say the curve γ to be innermost +if it bounds the disk Dγ ⊂ f k+(d2j−1) such that int Dγ contains no curves of +Γ. Consider this innermost curve γ. Due to Lemma 1 the surface ¯Σ0 \ ¯y0 is non- +compressible in R3 \ (O ∪ ¯Ln) and, therefore, there exists the disk dγ ⊂ ( ¯Σ0 \ ¯y0) +bounded by γ. Denote by Bγ ⊂ (R3 \(O∪ ¯Ln)) the 3-ball bounded by the 2-sphere +Dγ ∪ dγ. Consider two subcases: 1a) Bγ ⊂ QΣ0 and 1b) Bγ ⊂ (R3 \ int QΣ0). +In case 1a) let N(Bγ) ⊂ QΣ0 be a tubular neighborhood of the ball Bγ. Then +the set QΣ \ int N(Bγ) is h-compressible because the curve γ lies in its interior +and its boundary intersects ¯Ln at a unique point. The projection of this boundary + +Quasi-energy function for Pixton diffeomorphisms +13 +to S2 × S1 is, therefore, a secant surface of Ln of the same genus as Σ0. For it the +number of the connected components of the set Γ is less. One gets the same result +in case 1b) for the set QΣ0 ∪ N(Bγ). +If we continue this process then we get the secant surface of Ln of the same +genus as Σ0 and for which the set Γ contains no curves of type 1). Denote the +resulting surface again by Σ0. Now the set Γ consists only of the curves γ bounding +the disk Dγ ⊂ b2j−1 which contains the point b2j−1. Since (b2j−1 ⊔ c2j−1) ⊂ +(R3 \QΣ0), the number of these curves on the disk d2j−1 is even. Since the surface +¯Σ0\¯y0 is non-compressible in R3\(O∪ ¯Ln), all these curves are pairwise homotopic +on ¯Σ0 \ ¯y0 and, therefore, they lie in the annulus κ ⊂ ( ¯Σ0 \ ¯y0) bounded by the +pair of these curves γ1, γ2. Denote by ˜κ ⊂ d2j−1 the annulus bounded by the same +curves on the disk d2j−1. Let ˜Σ0 = ¯Σ0 \ κ ∪ ˜κ. Due to construction the surface +˜Σ0 is of the same genus as the surface ¯Σ0 and it bounds an h-compressible body. +Having removed a tubular neighborhood of the annulus ˜κ from this body we get a +h-compressible body whose boundary does not intersect the disk d2j−1 and whose +projection to S2 ×S1 is the secant surface of the knot Ln of the same genus as Σ0. +If we continue this process then we get a secant surface of Ln of the same +genus as Σ0 and for which the set Γ is not empty. Denote this surface again by +Σ0. Without loss of generality let the intersections of the surface ¯Σ0 with the +spheres f k(S2) be transversal. Denote by F the set of the connected components +of the intersection f k+(K) ∩ ¯Σ0. Now we show the way to reduce by 1 the number +of the components in F using isotopy of the secant surface. +Denote by Q the set obtained by removal from the annulus f k+(K) of the +tubular neighborhoods of the disks d2j−1 as well as the tubular neighborhoods of +the curves A2j, j ∈ {1, . . . , n}. Then Q is homeomorphic to the direct product +of the 2-sphere with 2n + 1 deleted points and the segment. Since Q ∩ ¯Σ0 = +f k+(K)∩ ¯Σ0 and since ¯Σ0 \ ¯y0 is non-compressible in R3 \(O∪ ¯Ln), each connected +component of F ∈ F is non-compressible in Q. Due to [16, Corollary 3.2] there +exists a surface ˜F ⊂ f k+−1(S2) diffeomorphic to F for which ∂F = ∂ ˜F and the +surface F ∪ ˜F bounds in Q the body ∆ diffeomorphic to the direct product F ×[0, 1]. +Then we replace the part F of ¯Σ0 with ˜F. If we continue the process we get the +desired secant surface Σ ⊂ K. +Notice (see, for instance, [4, Exercise 2.8.1]) that the fundamental group π1(K\ +¯Ln) has 2n generators γ1, . . . , γ2n, each of which γi, i ∈ {1, . . . , 2n} being the +generator of the punctured disk di \bi (Fig. 10). Since b2j−1 ∈ int QΣ and c2j−1 ∩ +QΣ = ∅, there exists the connected component of ˜d2j−1 of the intersection d2j−1 ∩ +QΣ which contains the point b2j−1. This component is the 2-disk bounded by the +curve ˜γ2j−1 ⊂ ( ¯Σ \ h(α1)) with holes and the curves γ2j−1, ˜γ2j−1 are homotopic +on the punctured disk d2j−1 \ b2j−1. In the same way one finds the curves ˜γ2j ⊂ +( ¯Σ \ h(α1)) homotopic to the curves γ2j on the punctured disk d2j \ b2j (Fig. 10). +Due to Lemma 1 the surface ¯Σ \ h(α1) is non-compressible in K \ ¯Ln. Then the +curves ˜γ1, . . . , ˜γ2n are pairwise non-homotopic to the generators on the surface +¯Σ \ h(α1). Therefore, the genus of the surface ¯Σ cannot be less than n. + +14 +Timur Medvedev, Olga Pochinka +Fig. 10 Generators of the group π1(K \ ¯Ln) +5 Construction of a quasi-energy function for a Pixton diffeomorphism +with the Hopf knot Ln +Let f be a Pixton diffeomorphism constructed for a generalized Mazur knot Ln. +Then its non-wandering set Ωf consists of four points: two sinks ω, S, a source N +and a saddle σ. Then W u +σ \σ consists of two separatrices ℓω, ℓS respective closures +of which contain the sinks ω, S, the separatrice ℓω being tame while ℓS being wild. +Let ¯Σ be the surface of genus n bounding the handle-body QΣ of the same genus. +Now we construct for f a Morse-Lyapunov function with 6 + 2n critical points. +Our construction of a quasi-energy function is analogous to the construction +of an energy function in [7]. +1. Choose an energy function ϕp : Up → R in the neighborhood of each fixed +point p of f so that ϕp(p) = dim W u +p . Let Bω, BS be the 3-balls which are the +level sets of respective functions ϕω, ϕS and such that BS ⊂ int QΣ. Choose a +tubular neighborhood Tσ of the arc W u +σ \ (Bω ∪ QΣ) so that the handle-body +Bω ∪ QΣ ∪ Tσ of genus n is f-compressible and its intersection with W s +σ is +the 2-disk. Denote by P + the smoothing of this body by addition of a small +exterior collar. +2. Due to [7, Section 4.3] there exists an energy function ϕ : P + \ int QΣ whose +value on ∂P + is 4/3, whose value on ¯Σ is 2/3 and which has exactly two critical +points ω, σ of respective Morse indexes 0, 1. The disks d1, . . . , d2n−1 cut the +handle-body QΣ making the 3-ball. Denote by BΣ the smoothing of this ball +by removal of the interior collar. The results of the classic Morse theory (see, +for example, [13]) allow to extend the function ϕ to the set QΣ \ int BΣ in +such way that it has n critical points of Morse index 1, one point lying on +each disk d1, . . . , d2n−1, while the value of ϕ on ∂BΣ is 1/3. Due to [7, Lemma + +Quasi-energy function for Pixton diffeomorphisms +15 +4.2] the function ϕ can be extended to the ball BΣ by an energy function +with the unique critical point S of Morse index 0. Since f(QΣ) ⊂ int BΣ, the +constructed function decreases along the trajectories of the diffeomorphism f. +3. It follows from the definition of the knot Ln that P − = S3 \ int P + is the +handle-body of genus n. Moreover, the disks d2, . . . , d2n cut P − making the +3-ball. Denote by B− smoothing of this ball by removal of the interior collar. +The results of the classic Morse theory (see, for example, [13]) allow extension +of the function ϕ to the set P − \ int B− in such way that it has n critical +points of Morse index 2, one point lying on each disk d2, . . . , d2n, while the +value of ϕ on ∂B− is 5/3. According to [7, Lemma 4.2] the function ϕ can be +extended to the ball B− by an energy function with unique critical point N +of Morse index 3. Since f(B−) ⊂ int P −, the constructed function decreases +along the trajectories of the diffeomorphism f and, therefore, it is the desired +quasi-energy function. +Conflict of interest +The authors declare that they have no conflict of interest. +References +1. Akhmetiev, P., Medvedev, T., Pochinka, O.: On the number of the classes of topological +conjugacy of Pixton diffeomorphisms. Qualitative Theory of Dynamical Systems 20(3), +1–15 (2021) +2. Bonatti, C., Grines, V.: Knots as topological invariants for gradient-like diffeomorphisms +of the sphere S3. Journal of Dynamical and Control Systems 6(4), 579–602 (2000) +3. Conley, C.: Isolated invariant sets and the morse index. American Mathematical Society, +CBMS, Providence, RI 38 (1978) +4. Daverman, R.J., Venema, G.: Embeddings in manifolds, vol. 106. American Mathematical +Soc. (2009) +5. Fomenko, A.: Differential Geometry and Topology: Additional Chapters. Moscow Univer- +sity Press (1983) +6. Grines, V., Laudenbach, F., Pochinka, O.: The energy function for gradient-like diffeomor- +phisms on 3-manifolds. Doklady Mathematics 78(2), 702–704 (2008) +7. Grines, V.Z., Laudenbach, F., Pochinka, O.V.: Quasi-energy function for diffeomorphisms +with wild separatrices. Mathematical Notes 86(1), 163–170 (2009) +8. Grines, V.Z., Medvedev, T.V., Pochinka, O.V.: Dynamical Systems on 2- and 3-Manifolds, +Developments in Mathematics, vol. 46. Springer International Publishing (2016). DOI +10.1007/978-3-319-44847-3 +9. Kirk, P., Livingston, C.: Knot invariants in 3-manifolds and essential tori. Pacific Journal +of Mathematics 197(1), 73–96 (2001) +10. Mazur, B.: A note on some contractible 4-manifolds. Annals of Mathematics 79(1), 221– +228 (1961) +11. Medvedev, T.V., Pochinka, O.V.: The wild Fox-Artin arc in invariant sets of dynamical +systems. Dynamical Systems 33(4), 660–666 (2018). DOI 10.1080/14689367.2017.1421903. +URL https://doi.org/10.1080/14689367.2017.1421903 +12. 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Annals of Math- +ematics pp. 56–88 (1968) + diff --git a/8dE0T4oBgHgl3EQffgB5/content/tmp_files/load_file.txt b/8dE0T4oBgHgl3EQffgB5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..33b47712682f88e4d7f73aa077fd131387ed1c0d --- /dev/null +++ b/8dE0T4oBgHgl3EQffgB5/content/tmp_files/load_file.txt @@ -0,0 +1,477 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf,len=476 +page_content='⋆ manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' (will be inserted by the editor) A quasi-energy function for Pixton diffeomorphisms defined by generalized Mazur knots Timur Medvedev · Olga Pochinka Received: date / Accepted: date Abstract In this paper we give a lower estimate for the number of critical points of the Lyapunov function for Pixton diffeomorphisms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Morse-Smale diffeo- morphisms in dimension 3 whose chain recurrent set consists of four points: one source, one saddle and two sinks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Bonatti and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Grines proved that the class of topological equivalence of such diffeomorphism f is completely defined by the equivalency class of the Hopf knot Lf that is the knot in the generating class of the fundamental group of the manifold S2 × S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' They also proved that there are infinitely many such classes and that any Hopf knot can be realized by a Pixton diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Pixton proved that diffeomorphisms defined by the standard Hopf knot L0 = {s} × S1 have an energy function (Lyapunov function) whose set of critical points coincide with the chain recurrent set whereas the set of critical points of any Lyapunov function for Pixton diffeomorphism with nontrivial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' non equivalent to the standard) Hopf knot is strictly larger than the chain recur- rent set of the diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The Lyapunov function for Pixton diffeomorphism with minimal number of critical points is called the quasi-energy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' In this paper we construct a quasi-energy function for Pixton diffeomorphisms defined by a generalized Mazur knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Keywords Hopf knot · Mazur knot · Pixton diffeomorphism · quasi-energy function Mathematics Subject Classification (2020) 37C15 · 37D15 ⋆ The research was done with the support of Russian National Foundation (project 21-11- 00010) except construction of the quasi-energy function which was supported by International Laboratory of Dynamical Systems and Applications of National Research University Higher School of Economics, grant of Government of Russian Federation 075-15-2022-1101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Medvedev Laboratory of Algorithms and Technologies for Network Analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' HSE University 136 Rodionova Street, Niznhy Novgorod, Russia E-mail: mtv2001@mail O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Pochinka International Laboratory of Dynamical Systems and Applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' HSE University, 25/12 Bol- shaya Pecherckaya Street, Niznhy Novgorod, Russia arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='02405v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='DS] 6 Jan 2023 2 Timur Medvedev, Olga Pochinka 1 Introduction and the main results Let M n be a smooth closed n-manifold with a metric d and let f : M n → M n be a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' For two given points x, y ∈ M n a sequence of points x = x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , xm = y is called an ε-chain of length m ∈ N connecting x to y if d(f(xi−1), xi) < ε for 1 ⩽ i ⩽ m (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' x x = 0 f x( ) 0 f x( ) 1 f x( ) 2 f x( ) m-1 f x( ) m 2 x1 x2 x3 xm-1 y x = m � � � � � Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 1 An ε-chain of length m ∈ N A point x ∈ M n is called chain recurrent for the diffeomorphism f if for every ε > 0 there is an ε-chain of length m connecting x to itself for some m (m depends on ε > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The chain recurrent set, denoted by Rf, is the set of all chain recurrent points of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Define the equivalence on Rf by the rule: x ∼ y if for every ε > 0 there is are ε-chains connecting x to y and y to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' This equivalence relation defines equivalence classes called chain components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' If the chain recurrent set of a diffeomorphism f is finite then it consists of periodic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' A periodic point p ∈ Rf of period mp is said to be hyperbolic if absolute values of all the eigenvalues of the Jacobian matrix � ∂f mp ∂x � |p are not equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' If absolute values of all these eigenvalues are greater (less) than 1 then p is called a sink (a source).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Sinks and sources are called knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' If a periodic point is not a knot then it is called a saddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Let p be a hyperbolic periodic point of a diffeomorphism f whose chain recur- rent set is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The Morse index of p, denoted by λp, is the number of eigen- values of Jacobian matrix whose absolute values are greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The stable manifold W s p = {x ∈ M n : lim k→+∞ d(f kmp(x), p) = 0} and the unstable manifold W u p = {x ∈ M n : lim k→+∞ d(f −kmp(x), p) = 0} of p are smooth manifolds diffeo- morphic to Rλp and Rn−λp, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Stable and unstable manifolds are called invariant manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' A connected component of the set W u p \\ p (W s p \\ p) is called a unstable (stable) separatrice of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' A diffeomorphism f : M n → M n is called a Morse-Smale diffeomorphism if 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' its chain recurrent set Rf consists of finite number of hyperbolic points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' for any two points p, q ∈ Rf the manifolds W s p , W u q intersect transversally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Quasi-energy function for Pixton diffeomorphisms 3 C Conley in [3] gave the following definition: a Lyapunov function for a Morse- Smale diffeomorphism f : M n → M n is a continuous function ϕ : M n → R satisfying – ϕ(f(x)) < ϕ(x) if x /∈ Rf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' – ϕ(f(x)) = ϕ(x) if x ∈ Rf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Notice that every Morse-Smale diffeomorphism f has a Morse-Lyapunov func- tion 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' a Lyapunov function ϕ : M n → R which is a Morse function such that each periodic point p ∈ Rf is its non-degenerate critical point of index λp with Morse coordinates (Vp, φp : y ∈ Vp �→ (x1(y), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , xn(y)) ∈ Rn and φ−1 p (Ox1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' xλp) ⊂ W u p , φ−1 p (Oxλp+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' xn) ⊂ W s p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' (∗) If the function ϕ has no critical points outside Rf then following [15] we call it the energy function for the Morse-Smale diffeomorphism f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The proof of existence of an energy Morse function for a Morse-Smale diffeo- morphism of the circle is an easy exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Pixton [15] in 1977 proved that every Morse-Smale diffeomorphism of a surface has an energy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' There he also constructed an example of a Morse-Smale diffeomorphism on the 3-sphere which admits no energy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The obstacle to existence of an energy function in his example was the wild embedding of the saddle separatrices in the ambient manifold (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' the closure of the separatrice is not a submanifold of the ambient space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' From [11] it follows that there are Morse-Smale diffeomorphisms with no energy function on manifolds of any dimension n > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Therefore, following [7] for a Morse-Smale diffeomorphism f we call a Morse-Lyapunov function with the minimal number of critical points (denote it by ρf ) a quasi-energy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Notice that ρf is a topological invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' if two diffeomorphisms f, f ′ : M n → M n are topologically conjugate (that is there is a diffeomorphism h : M n → M n such that h ◦ f = f ′ ◦ h) then ρf = ρf′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' In this paper we give a lower estimate of ρf for Pixton diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The class of Pixton diffeomorphisms P is defined in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Every diffeo- morphism f ∈ P is a Morse-Smale 3-diffeomorphism whose chain recurrent set consists of four points: one source, one saddle and two sinks (for details see section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Notice that Pixton’s example is a diffeomorphism of this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' According to [2] the class of topological conjugacy of a diffeomorphism f ∈ P is completely defined by the equivalence class of the Hopf knot Lf, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' the knot in the generating class of the fundamental group of the manifold S2 × S1 (see Proposition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Moreover, any Hopf knot can be realized as a Pixton diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Recall that a knot in S2 × S1 is a smooth embedding γ : S1 → S2 × S1 or the image of this embedding L = γ(S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Two knots L, L′ are said to be equivalent if there is a homeomorphism h : S2 × S1 → S2 × S1 such that h(L) = L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Two knots 1 This function can be constructed, for example, by suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Consider the topological flow ˆft on the manifold Mn × R defined by ˆft(x) = x + t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Define the diffeomorphism g : Mn × R → Mn × R by g(x, τ) = (f(x), τ − 1) and let G = {gk , k ∈ Z} and W = (Mn × R)/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Denote by pW : Mn × R → W the natural projection and denote by ft the flow on W defined by ft(x) = pW ( ˆft(p−1 W (x))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The flow ft is called the suspension over f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' By construction the chain recurrent set of ft consists of the finite number of periodic orbits δi = pW (Oi × R), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , kf} and this means that the suspension ft is a Morse-Smale flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' A Lyapunov function for these flows is constructed in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then the restriction of this function on M is the desired Lyapunov function for f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 4 Timur Medvedev, Olga Pochinka γ, γ′ are smoothly homotopic if there exists a smooth map Γ : S1 ×[0, 1] → S2 ×S1 such that Γ(s, 0) = γ(s) and Γ(s, 1) = γ′(s) for every s ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' If Γ|S1×{t} is an embedding for every t ∈ [0, 1] then the knots are said to be isotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Any Hopf knot L ⊂ S2 × S1 is smoothly homotopic to the standard Hopf knot L0 = {s} × S1 (see, for example, [9]) but generally it is neither isotopic nor equivalent to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Mazur [10] constructed the Hopf knot LM which we call the Mazur knot and which is non-equivalent and non-isotopic to L0 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' It follows from the results of [1] that there exists a countable family of pairwise Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 2 Two non-isotopic and non equivalent Hopf knots L0 and LM: a) the standard Hopf knot L0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' b) the Mazur knot LM non-equivalent Hopf knots Ln, n ∈ N which are generalized Mazur knots (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 3 A generalized Mazur knot Ln According to [6] a Pixton diffeomorphism f admits an energy Morse function if and only if the knot Lf is trivial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' equivalent to the standard one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' If the Quasi-energy function for Pixton diffeomorphisms 5 knot Lf is not trivial then the number ρf of the critical points of a quasi-energy Morse function of f is evidently even and ρf ⩾ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The main result of this paper is the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Theorem 1 Let f be a Pixton diffeomorphism (f ∈ P) and let Ln, n ∈ N be its knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then the number ρf of critical points of a quasi-energy function of f is calculated by2 ρf = 4 + 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 2 Construction of Pixton diffeomorphisms In dynamics a wild Artin-Fox arc was for the first time introduced by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Pixton in [15] where he constructed a Morse-Smale diffeomorphism on the 3-sphere with the unique saddle whose invariant manifolds form an Artin-Fox arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' We give the modern construction of these diffeomorphisms following Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Bonatti and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Grines [2] where Pixton diffeomorphisms were also classified (see also [8], [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' For x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , xn) ∈ Rn denote ||x|| = � x2 1 + · · · + x2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Let h : R3 → R3 be the diffeomorphism defined by h(x1, x2, x3) = � x1 2 , x2 2 , x3 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Define the map p : R3 \\ O → S2 × S1 by p(x1, x2, x3) = � x1 ||x||, x2 ||x||, log2(||x||) (mod 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Let L ⊂ (S2 ×S1) be a Hopf knot and let U(L) be its tubular neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then the set ¯L = p−1(L) is the h-invariant arc in R3 and U(¯L) = p−1(U(L)) is its h-invariant neighborhood diffeomorphic to D2 × R1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Let C = {(x1, x2, x3) ∈ R3 : x2 2 + x2 3 ⩽ 4} and let gt : C → C be the flow defined by gt(x1, x2, x3) = (x1 + t, x2, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then there is a diffeomorphism ζ : U(L) → C that conjugates h|U(L) and g = g1|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Define the flow φt on C by: � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ˙x1 = � 1 − 1 9(x2 1 + x2 2 + x2 3 − 4)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' x2 1 + x2 2 + x2 3 ⩽ 4 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' x2 1 + x2 2 + x2 3 > 4 ˙x2 = � � � � � x2 2 � sin � π 2 � x2 1 + x2 2 + x2 3 − 3 �� − 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 2 < x2 1 + x2 2 + x2 3 ⩽ 4 −x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' x2 1 + x2 2 + x2 3 ⩽ 2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' x2 1 + x2 2 + x2 3 > 4 ˙x3 = � � � � � x3 2 � sin � π 2 � x2 1 + x2 2 + x2 3 − 3 �� − 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 2 < x2 1 + x2 2 + x2 3 ⩽ 4 −x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' x2 1 + x2 2 + x2 3 ⩽ 2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' x2 1 + x2 2 + x2 3 > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' By construction the diffeomorphism φ = φ1 has two fixed points: the saddle P(1, 0, 0) and the sink Q(−1, 0, 0) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 5), both being hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' One unstable 2 For n = 1 Theorem 1 is proved in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 6 Timur Medvedev, Olga Pochinka L L _ U(L) _ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 4 Suspension of a Hopf knot 1 1 2 O Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 5 Trajectories of the flow φt separatrice of the saddle P coincides with the open interval � (x1, x2, x3) ∈ R3 : |x1| < 1, x2 = x3 = 0 � in the basin of the sink Q while the other coincides with the ray � (x1, x2, x3) ∈ R3 : x1 > 1, x2 = x3 = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Notice that φ coincides with the diffeomorphism g = g1 outside the ball {(x1, x2, x3) ∈ C : x2 1 + x2 2 + x2 3 ⩽ 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Define the diffeomorphism ¯fL : R3 → R3 so that ¯fL coincides with h outside U(L) and it coincides with ζ−1φζ on U(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then ¯fL has in U(L) two fixed points: the sink ζ−1(Q) and the saddle ζ−1(P), both being hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The unstable separatrice of the saddle ζ−1(P) lies in L (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Now project the dynamics onto the 3-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Denote by N(0, 0, 0, 1) the North Pole of the sphere S3 = {x = (x1, x2, x3, x4) : ||x|| = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' For every point x ∈ (S3 \\ {N}) there is the unique line passing through N and x in R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' This line intersects Quasi-energy function for Pixton diffeomorphisms 7 R Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 6 The phase portrait of the diffeomorphism ¯fL R3 in exactly one point ϑ+(x) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The point ϑ+(x) is the stereographic projection of the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' One can easily check that ϑ+(x1, x2, x3, x4) = � x1 1 − x4 , x2 1 − x4 , x3 1 − x4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Thus, the stereographic projection ϑ+ : S3 \\ {N} → R3 is a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' N x ��(x) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 7 The stereographic projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' By construction ¯fL coincides with h in some neighborhood of the point O and in some neighborhood of the infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Therefore, it induces on S3 the Morse-Smale fL(x) = � ϑ−1 + ( ¯fL(ϑ+(x))), x ̸= N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' N, x = N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' It follows directly from the construction that the non-wandering set of fL consists of exactly four fixed hyperbolic points: two sinks ω = ϑ−1 + (ζ−1(Q)), S, one saddle σ = ϑ−1 + (ζ−1(P)) and one source N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' We say the constructed diffeomorphism to be model and it is of Pixton class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 8 Timur Medvedev, Olga Pochinka Proposition 1 ([2]) – Any diffeomorphism f ∈ P is topologically conjugate to some model diffeomor- phism fL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' – Two model diffeomorphisms fL, fL′ are topologically conjugate if and only if their knots L, L′ are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 3 Genus of Hopf knot In this section we introduce the notion of genus for a Hopf knot and use it to estimate the number of critical points of the quasi-energy function of the Pixton diffeomorphism defined by this knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Let L be a Hopf knot and let ¯L = p−1(L) be its cover in R3\\O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' We say a closed orientable surface Σ ⊂ S2 × S1 to be a secant surface of the knot L if it intersects L in a unique point and there is an h-compressible 3-manifold QΣ ⊂ R3 (that is h(QΣ) ⊂ int QΣ) with the boundary ¯Σ such that Σ = p( ¯Σ) and the intersection ¯L∩ ¯Σ is the unique point ¯y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The minimally possible genus gL of the secant surface is called the genus of the knot L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The secant surface of L of genus gL is said to be minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Lemma 1 If Σ is a minimal secant surface of the knot L then the surface ¯Σ \\ ¯y is non-compressible in R3 \\ (O ∪ ¯L), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' any simple closed curve c ⊂ int ( ¯Σ \\ ¯y) is contractible on ¯Σ\\ ¯y if it bounds a smoothly embedded 2-disk D ⊂ int (R3\\(O∪ ¯L)) such that D ∩ ( ¯Σ \\ ¯y) = ∂D = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Proof Let Σ be a minimal secant surface of L and let ¯y be the unique point of the intersection ¯L ∩ ¯Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Assume the opposite: the surface ¯Σ \\ ¯y is compressible in R3 \\ (O ∪ ¯L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then there is a non-contractible simple closed curve c ⊂ int ( ¯Σ \\ ¯y) and there is the smoothly embedded 2-disk D ⊂ int (R3 \\ (O ∪ ¯L)) such that D ∩ ( ¯Σ \\ ¯y) = ∂D = c (see, for example, [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then we have two possibilities: (int D) ∩ � � k∈Z hk( ¯Σ) � = ∅, (1) (int D) ∩ � � k∈Z hk( ¯Σ) � ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' (2) In case (1) two subcases are possible: (1a) D ⊂ QΣ, (1b) D ⊂ (R3 \\ int QΣ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' For case 1a) let N(D) ⊂ QΣ be a tubular neighborhood of the disk D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then exactly one connected component of the set QΣ \\ int N(D) intersects ¯L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' According to (1) this component is h-compressible and its boundary intersects ¯L at a unique point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The projection of this boundary into S2 × S1 is, therefore, the secant surface of L of genus less than gL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' This contradicts the fact that the surface Σ is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' In case 1b) let N(D) ⊂ (R3 \\ int QΣ) be a tubular neighborhood of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then due to (1) the set QΣ ∪ N(D) is h-compressible and its boundary intersects ¯L at a unique point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The projection of this boundary into S2 ×S1 is, therefore, the secant surface of L of genus less than gL and we have the same contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Quasi-energy function for Pixton diffeomorphisms 9 In case (2) without loss of generality assume the intersection int D∩( � k∈Z hk( ¯Σ)) to be transversal and denote it by Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Let γ be a curve from Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' We say the curve γ to be innermost if it is the boundary of the disk Dγ ⊂ D such that int Dγ contains no curves of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Consider this innermost curve γ ⊂ f k(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' There are two subcases: a) γ is essential on f k(Σ) and b) γ is contractible on f k(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' In case a) the arguments of the case (1) apply for the body f k(QΣ) and the disk Dγ and we get the contradiction to the minimality of the surface Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' In case b) denote by dγ ⊂ f k(Σ) the 2-disk bounded by γ and denote by Bγ ⊂ (R3 \\ O) the 3-ball bounded by the 2-sphere Dγ ∪ dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Consider: b1) Bγ ⊂ f k(QΣ) and b2) Bγ ⊂ (R3 \\ int f k(QΣ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' For b1) let N(Bγ) ⊂ f k(QΣ) be a tubular neighborhood of Bγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then the set QΣ \\ int N(Bγ) is h-compressible because the curve γ lies in its interior and the boundary of QΣ \\ int N(Bγ) intersects ¯L at a unique point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The projection of this boundary into S2 ×S1 is, therefore, the secant surface of the knot L of genus gL for which the number of connected components of the set Γ is less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' We get the same result for b2) for the set QΣ ∪ N(Bγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Thus, iterating the process we come either to the case a) or to the case (1) and get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Lemma 2 For any diffeomorphism f ∈ P the following estimation holds ρf ⩾ 4 + 2gLf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' (3) Proof Since Proposition 1 is true and since the number ρf of the critical points of a quasi-energy function of f ∈ P is invariant, from now on we consider model Pixton diffeomorphismsfL with the Hopf knot L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Denote by ℓ the non-stable separatrice of the saddle σ lying in the basin of the sink S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Let pS : W s S \\ S → S2 × S1 be the natural projection sending a point w ∈ (W s S\\S) to the point p(f kw(w)), f kw(w) ∈ VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Since the diffeomorphism fL coincides with the homothety h in some neigh- borhood VS of S, the natural projection pS is well defined and pS(ℓ) = L by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Consider an arbitrary Morse-Lyapunov function ϕ : S3 → R of the diffeomor- phism fL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' To be definite let ϕ(S) = 0, ϕ(σ) = 1 and ϕ(N) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' From the definition of the Morse-Lyapunov function it follows that ϕ|ℓ monotonically decreases in some neighborhood of the saddle σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Therefore, there is ε1 ∈ (0, 1) such that the interval (1 − ε1, 1) contains no critical values of ϕ and the connected component ¯Σ1 of the level set ϕ−1(1 − ε1) intersects the separatrice ℓ at the unique point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Denote this point by w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Let ¯Q1 be the connected component of the set ϕ−1([0, 1 − ε1]) which contains the segment [w1, S] of the closure of the separatrice ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Since ϕ decreases along the trajectories of f, the values of ϕ on W s σ are greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Therefore, the manifold ¯Q1 lies in the manifold W s S diffeomorphic to R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Let the function ϕ| ¯ Q1 have kq, q ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , 3} critical points of index q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Due to [5, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='1] on the manifold ¯Q1 there exists a self-indexing Morse function ψ (the value of the function in a critical point equals the index of this point) which has kq critical points of index q and which is constant on ∂ ¯Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Thus, the manifold ¯Q1 is the surface ˜Q1 of 10 Timur Medvedev, Olga Pochinka genus g1 = 1 + k1 − k0 with attached handles of indexes 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then the genus of any surface of the set ∂ ¯Q1 cannot be greater than g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' On the other hand, the number of critical points of ϕ| ¯ Q1 is not less than k0+k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' If k0 ⩾ 1 and g1 = 1 + k1 − k0 then one gets k0 + k1 = g1 + 2k0 − 1 ⩾ g1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Thus, ϕ| ¯ Q1 has at least g1 + 1 critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Denote by ¯Σ1 the connected component of ∂ ¯Q1 which intersects the separatrice ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then the surface ¯Σ1 divides the manifold W s S ∼= R3 into two parts, one of which Q1 being an h-compressible body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' This means that Σ1 = pS( ¯Σ1) is the secant surface of L and, therefore, g1 ⩾ gL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Analogously, there is ε2 ∈ (0, 1) for which the interval (1, 1 + ε2) contains no critical points of ϕ and the connected component ¯Q2 of the level set ϕ−1([0, 1+ε2)] contains cl(W u σ ) in its interior while the intersection ¯Q2 with W s σ is the unique 2-disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Due to construction the function ϕ| ¯ Q2 has at least g1 + 3 critical points and genus of the connected components of ∂ ¯Q2 is less or equals g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Denote by ¯Σ2 the connected component of ∂ ¯Q2 which intersects W s σ and denote by g2 its genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The surface ¯Σ2 divides the manifold W u N ∼= R3 into two parts, one of which Q2 being an h−1-compressible body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Arguing as above one comes to conclusion that the number of critical points of ϕ|Q2 is at least g2 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Therefore, the total number of critical points of ϕ is greater or equal to g1 + 3 + g2 + 1 ⩾ 4 + 2g1 ⩾ 4 + 2gLf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 4 The generalized Mazur knot Ln In this section we show that the genus gLn of a generalized Mazur knot equals n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' At first we give a detailed description of construction of Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='1 Construction of the generalized Mazur knot Ln Recall that h : R3 → R3 is the homothety defined by h(x1, x2, x3) = �x1 2 , x2 2 , x3 2 � and p : R3 \\ O → S2 × S1 is the natural projection defined by p(x1, x2, x3) = � x1 ||x||, x2 ||x||, log2(||x||) (mod 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Consider the annulus K = � (x1, x2, x3) ∈ R3 : 1 4 ≤ x2 1 + x2 2 + x2 3 ≤ 1 � bounded by the spheres S2 = � (x1, x2, x3) ∈ R3 : x2 1 + x2 2 + x2 3 = 1 � , h(S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Quasi-energy function for Pixton diffeomorphisms 11 Pick on the circle S1 = � (x1, x2, x3) ∈ R3 : x2 1 + x2 2 = 1, x3 = 0 � pairwise distinct points α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , α2n+1 numbered in counter-clockwise order (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Let ai, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , 2n} be the arc of the circle S1 bounded by αi, αi+1 whose inte- rior contains no points of {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , α2n+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Let B, Ai ⊂ int K, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , 2n} be pairwise disjoint smooth arcs such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' the boundary points of B are α2n+1, h(α1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' the boundary points of A2j−1 are α2j−1, α2j and the boundary points of A2j are h(α2j), h(α2j+1) for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' the closed curves c2j−1 = cl(a2j−1 ∪ A2j−1), c2j = cl(h(a2j) ∪ A2j) bound the 2-disks d2j−1, d2j, the transversal intersection of these disks being the arc lj with the boundary points b2j−1 = d2j−1 ∩ A2j and b2j = d2j ∩ A2j−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' the arc cl(h(A1) ∪ A2 ∪ · · · ∪ h(A2n−1) ∪ A2n ∪ B) is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 8 Construction of the knot Ln Let ¯Ln = � k∈Z hk(B ∪ A1 ∪ · · · ∪ A2n), Ln = p(¯Ln).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='2 The genus of the knot Ln Lemma 3 The genus gLn of the knot Ln equals n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 12 Timur Medvedev, Olga Pochinka Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 9 A secant surface of Ln of genus n Proof Since there is a secant surface of Ln of genus n, we have gLn ⩽ n (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Now we show that gLn ⩾ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' To that end we prove that for Ln there exists a minimal secant surface Σ such that ¯Σ ⊂ K and ¯Ln ∩ ¯Σ = h(α1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Indeed, let Σ0 be some minimal secant surface of Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then there exists the connected component ¯Σ0 of p−1(Σ0) such that it intersects the curve ¯Ln at the point ¯y0 situated on ¯Ln between α1, h(α1) and that bounds the h-compressible body QΣ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Without loss of generality let ¯y0 = h(α1) (otherwise the desired surface is constructed by removing the tubular neighborhood of the arc [¯y0, h(α1)] ⊂ ¯Ln from QΣ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Denote by k+, k− ⩾ 0 the maximal integers for whichf k( ¯Σ0)∩ ¯Σ0 ̸= ∅, f −k( ¯Σ0)∩ ¯Σ0 ̸= ∅, k ⩾ 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' If k+ = k− = 0 then ¯Σ0 is the desired surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Other- wise we show the way to decrease by 1 the number k+ > 0 (for k− the arguments are the same) using isotopy of the secant surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Notice that ¯Σ0 ∩ f k+(c2j−1) = ∅, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Without loss of generality let the intersection Γ = n� j=1 f k+(d2j−1) ∩ ¯Σ0 be transversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Let γ be a curve from Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then γ bounds the unique disk Dγ ⊂ f k+(d2j−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' There are two possibilities: 1) b2j−1 /∈ Dγ, 2) b2j−1 ∈ Dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' In case 1) we say the curve γ to be innermost if it bounds the disk Dγ ⊂ f k+(d2j−1) such that int Dγ contains no curves of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Consider this innermost curve γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Due to Lemma 1 the surface ¯Σ0 \\ ¯y0 is non- compressible in R3 \\ (O ∪ ¯Ln) and, therefore, there exists the disk dγ ⊂ ( ¯Σ0 \\ ¯y0) bounded by γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Denote by Bγ ⊂ (R3 \\(O∪ ¯Ln)) the 3-ball bounded by the 2-sphere Dγ ∪ dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Consider two subcases: 1a) Bγ ⊂ QΣ0 and 1b) Bγ ⊂ (R3 \\ int QΣ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' In case 1a) let N(Bγ) ⊂ QΣ0 be a tubular neighborhood of the ball Bγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then the set QΣ \\ int N(Bγ) is h-compressible because the curve γ lies in its interior and its boundary intersects ¯Ln at a unique point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The projection of this boundary Quasi-energy function for Pixton diffeomorphisms 13 to S2 × S1 is, therefore, a secant surface of Ln of the same genus as Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' For it the number of the connected components of the set Γ is less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' One gets the same result in case 1b) for the set QΣ0 ∪ N(Bγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' If we continue this process then we get the secant surface of Ln of the same genus as Σ0 and for which the set Γ contains no curves of type 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Denote the resulting surface again by Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Now the set Γ consists only of the curves γ bounding the disk Dγ ⊂ b2j−1 which contains the point b2j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Since (b2j−1 ⊔ c2j−1) ⊂ (R3 \\QΣ0), the number of these curves on the disk d2j−1 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Since the surface ¯Σ0\\¯y0 is non-compressible in R3\\(O∪ ¯Ln), all these curves are pairwise homotopic on ¯Σ0 \\ ¯y0 and, therefore, they lie in the annulus κ ⊂ ( ¯Σ0 \\ ¯y0) bounded by the pair of these curves γ1, γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Denote by ˜κ ⊂ d2j−1 the annulus bounded by the same curves on the disk d2j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Let ˜Σ0 = ¯Σ0 \\ κ ∪ ˜κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Due to construction the surface ˜Σ0 is of the same genus as the surface ¯Σ0 and it bounds an h-compressible body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Having removed a tubular neighborhood of the annulus ˜κ from this body we get a h-compressible body whose boundary does not intersect the disk d2j−1 and whose projection to S2 ×S1 is the secant surface of the knot Ln of the same genus as Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' If we continue this process then we get a secant surface of Ln of the same genus as Σ0 and for which the set Γ is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Denote this surface again by Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Without loss of generality let the intersections of the surface ¯Σ0 with the spheres f k(S2) be transversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Denote by F the set of the connected components of the intersection f k+(K) ∩ ¯Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Now we show the way to reduce by 1 the number of the components in F using isotopy of the secant surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Denote by Q the set obtained by removal from the annulus f k+(K) of the tubular neighborhoods of the disks d2j−1 as well as the tubular neighborhoods of the curves A2j, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then Q is homeomorphic to the direct product of the 2-sphere with 2n + 1 deleted points and the segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Since Q ∩ ¯Σ0 = f k+(K)∩ ¯Σ0 and since ¯Σ0 \\ ¯y0 is non-compressible in R3 \\(O∪ ¯Ln), each connected component of F ∈ F is non-compressible in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Due to [16, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='2] there exists a surface ˜F ⊂ f k+−1(S2) diffeomorphic to F for which ∂F = ∂ ˜F and the surface F ∪ ˜F bounds in Q the body ∆ diffeomorphic to the direct product F ×[0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then we replace the part F of ¯Σ0 with ˜F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' If we continue the process we get the desired secant surface Σ ⊂ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Notice (see, for instance, [4, Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='1]) that the fundamental group π1(K\\ ¯Ln) has 2n generators γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , γ2n, each of which γi, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , 2n} being the generator of the punctured disk di \\bi (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Since b2j−1 ∈ int QΣ and c2j−1 ∩ QΣ = ∅, there exists the connected component of ˜d2j−1 of the intersection d2j−1 ∩ QΣ which contains the point b2j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' This component is the 2-disk bounded by the curve ˜γ2j−1 ⊂ ( ¯Σ \\ h(α1)) with holes and the curves γ2j−1, ˜γ2j−1 are homotopic on the punctured disk d2j−1 \\ b2j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' In the same way one finds the curves ˜γ2j ⊂ ( ¯Σ \\ h(α1)) homotopic to the curves γ2j on the punctured disk d2j \\ b2j (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Due to Lemma 1 the surface ¯Σ \\ h(α1) is non-compressible in K \\ ¯Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then the curves ˜γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , ˜γ2n are pairwise non-homotopic to the generators on the surface ¯Σ \\ h(α1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Therefore, the genus of the surface ¯Σ cannot be less than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 14 Timur Medvedev, Olga Pochinka Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 10 Generators of the group π1(K \\ ¯Ln) 5 Construction of a quasi-energy function for a Pixton diffeomorphism with the Hopf knot Ln Let f be a Pixton diffeomorphism constructed for a generalized Mazur knot Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then its non-wandering set Ωf consists of four points: two sinks ω, S, a source N and a saddle σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Then W u σ \\σ consists of two separatrices ℓω, ℓS respective closures of which contain the sinks ω, S, the separatrice ℓω being tame while ℓS being wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Let ¯Σ be the surface of genus n bounding the handle-body QΣ of the same genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Now we construct for f a Morse-Lyapunov function with 6 + 2n critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Our construction of a quasi-energy function is analogous to the construction of an energy function in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Choose an energy function ϕp : Up → R in the neighborhood of each fixed point p of f so that ϕp(p) = dim W u p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Let Bω, BS be the 3-balls which are the level sets of respective functions ϕω, ϕS and such that BS ⊂ int QΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Choose a tubular neighborhood Tσ of the arc W u σ \\ (Bω ∪ QΣ) so that the handle-body Bω ∪ QΣ ∪ Tσ of genus n is f-compressible and its intersection with W s σ is the 2-disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Denote by P + the smoothing of this body by addition of a small exterior collar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Due to [7, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='3] there exists an energy function ϕ : P + \\ int QΣ whose value on ∂P + is 4/3, whose value on ¯Σ is 2/3 and which has exactly two critical points ω, σ of respective Morse indexes 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The disks d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , d2n−1 cut the handle-body QΣ making the 3-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Denote by BΣ the smoothing of this ball by removal of the interior collar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The results of the classic Morse theory (see, for example, [13]) allow to extend the function ϕ to the set QΣ \\ int BΣ in such way that it has n critical points of Morse index 1, one point lying on each disk d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , d2n−1, while the value of ϕ on ∂BΣ is 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Due to [7, Lemma Quasi-energy function for Pixton diffeomorphisms 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='2] the function ϕ can be extended to the ball BΣ by an energy function with the unique critical point S of Morse index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Since f(QΣ) ⊂ int BΣ, the constructed function decreases along the trajectories of the diffeomorphism f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' It follows from the definition of the knot Ln that P − = S3 \\ int P + is the handle-body of genus n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Moreover, the disks d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , d2n cut P − making the 3-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Denote by B− smoothing of this ball by removal of the interior collar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' The results of the classic Morse theory (see, for example, [13]) allow extension of the function ϕ to the set P − \\ int B− in such way that it has n critical points of Morse index 2, one point lying on each disk d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' , d2n, while the value of ϕ on ∂B− is 5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' According to [7, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content='2] the function ϕ can be extended to the ball B− by an energy function with unique critical point N of Morse index 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Since f(B−) ⊂ int P −, the constructed function decreases along the trajectories of the diffeomorphism f and, therefore, it is the desired quasi-energy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' Conflict of interest The authors declare that they have no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'} +page_content=' References 1.' metadata={'source': 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b/CdE0T4oBgHgl3EQfgQEc/content/tmp_files/2301.02414v1.pdf.txt @@ -0,0 +1,1101 @@ +Prepared for submission to JCAP +Stochastic gravitational wave +background from the collisions of dark +matter halos +Qiming Yana,b Xin Rena,b Yaqi Zhaoa,b Emmanuel N. Saridakisc,a,d +aDeep Space Exploration Laboratory/School of Physical Sciences, University of Science and +Technology of China, Hefei, Anhui 230026, China +bCAS Key Laboratory for Researches in Galaxies and Cosmology/Department of Astronomy, +School of Astronomy and Space Science, University of Science and Technology of China, +Hefei, Anhui 230026, China +cNational Observatory of Athens, Lofos Nymfon, 11852 Athens, Greece +dDepartamento de Matem´aticas, Universidad Cat´olica del Norte, Avda. +Angamos 0610, +Casilla 1280 Antofagasta, Chile +E-mail: asadoubi233@mail.ustc.edu.cn, rx76@mail.ustc.edu.cn, +zxmyg86400@mail.ustc.edu.cn, msaridak@noa.gr +Abstract. +We investigate for the first time the effect of the dark matter (DM) halos colli- +sions, namely collisions of galaxies and galaxy clusters, through gravitational bremsstrahlung, +on the stochastic gravitational wave background. We first calculate the gravitational wave +signal of a single collision event, assuming point masses and linear perturbation theory. Then +we proceed to the calculation of the energy spectrum of the collective effect of all dark matter +collisions in the Universe. Concerning the DM halo collision rate we show that it is given +by the product of the number density of DM halos, which is calculated by the extended +Press-Schechter (EPS) theory, with the collision rate of a single DM halo, which is given by +simulation results, with a function of the linear growth rate of matter density through cos- +mological evolution. Hence, integrating over all mass and distance ranges, we finally extract +the spectrum of the stochastic gravitational wave background created by DM halos collisions. +As we show, the resulting contribution to the stochastic gravitational wave background is +of the order of hc ≈ 10−30 in the pulsar timing array (PTA) band of f ≈ 10−9Hz, much +smaller than other GW sources, such as super-massive black-hole mergers. However, in very +low frequency band, it is larger. With current observational sensitivity it cannot be detected, +nevertheless it may be accessible by PTA in the future, where techniques of distinguishing +signal overlap should be used in order to isolate it and use it for cosmological studies. +arXiv:2301.02414v1 [astro-ph.CO] 6 Jan 2023 + +Contents +1 +Introduction +1 +2 +Gravitational waves emitted during a single collision +2 +3 +Effect on the stochastic gravitational wave background +6 +3.1 +Energy spectrum of a single GW event +7 +3.2 +Number density of GW sources +8 +3.3 +The energy spectrum of the stochastic gravitational wave background +10 +4 +Conclusions +11 +1 +Introduction +Recently, the gravitational wave (GW) detecting technology has been developing rapidly. +In 2015, the detection of binary black holes merger GW150914 by the LIGO experimental +cooperation signaled the first detection of gravitational waves [1], while in 2017, the joint +detection of GW170817 [2] and GRB170817A [3] opened the new era of multi-messenger +astronomy [4]. In general, with the increasing amount of detected gravitational wave events +[5] one has improved statistics that allows to track the history of the universe [6, 7] and +impose bounds on various cosmological parameters [8, 9], as well as constrain various theories +of gravity [10–14]. Moreover, for different frequencies and types of gravitational wave sources, +various detection means have been designed and implemented. Besides ground-based laser +interferometers such as LIGO, Virgo and KAGRA, which probe high frequency bands (10 − +104 Hz), space-based laser interferometers such as LISA [15, 16] for intermediate frequency +gravitational waves (10−4 − 1 Hz), and the pulsar timing array (PTA) [17–20] for lower +frequency bands (10−9 − 10−6 Hz), are also raised. These observational avenues allow us to +acquire rich information from GWs of different types and sources, among which stochastic +gravitational wave background is attracting increasing interest. +Stochastic gravitational wave background (GWB) is a type of random background signal +that exists in an analogous way to the cosmic microwave background. +The contribution +of GWB can be roughly divided into cosmological sources and astrophysical sources [21]. +Astrophysical originated GWB contains all types of unresolved GW emitting events, including +binary black hole mergers [22–28]. These signals can provide information about astrophysical +source populations and processes over the history of the universe [29–32]. On the other hand, +cosmological originated GWB mainly involves primordial gravitational perturbations during +the inflation epoch [33–35], or perturbations arising from primordial black holes fluctuations +[36–39]. GW signals typically remain unaffected during their propagation, and thus they +can provide valuable information about the very early stages of the universe. For instance, +different inflationary models can lead to different predictions for the GWB spectrum [40–52], +and thus GWB can be used as a probe of this primordial universe epoch. Since GWB can +provide us with important astrophysical and cosmological probes, it is crucial to understand +its composition and properties [53–63]. +On the other hand, according to observations, dark matter (DM) constitutes a significant +fraction of the energy density of the universe [64–66]. Its microphysical nature and possible +– 1 – + +interactions remain unknown [67–70], nevertheless we do know unambiguously that DM +interacts gravitationally [71, 72]. +Current theory predicts that the main part of DM is +concentrated in dark halos, which coincide in position with galaxy or galaxy clusters [73]. +These galaxies and galaxy clusters, and thus dark halos too, are typically accelerating and +merging through their mutual attraction [74–76]. Such processes can in principle release GW +signal through gravitational bremsstrahlung [77–87]. +In this work, we are interested in investigating for the first time the possible GW signals +that could be emitted through bremsstrahlung during dark halo merger and collisions, and +their contribution to the stochastic GWB. In particular, we will first consider a single event of +two DM halos collision, and we will calculate the emitted GW signal. Then, we will calculate +the energy spectrum contribution to the stochastic GWB, taking the DM halo collision rate +into consideration. The structure of the article is as follows. In Section 2 we analyze the +GW emitted during the collision of two galaxies or two galaxy clusters. In Section 3 we +integrate over redshift and DM halos parameters to extract the contribution to stochastic +GWB. Finally, in Section 4 we conclude and discuss our results. +2 +Gravitational waves emitted during a single collision +In this section, we aim at estimating the gravitational waves emitted during a single collision +event. In particular, we calculate the GW radiated by the collision of two DM halos, which +corresponds to the collision of two galaxies or two galaxy clusters. +According to observations, such a collision typically has a huge duration, which in turn +implies that the energy radiated through GWs per unit time is not very large, and thus we +can safely use linear perturbation theory in the involved calculations. Specifically, we use +[88] +gµν = ηµν + hµν, +|hµν| ≪ 1, +(2.1) +¯hij(t, x) = 2G +rc4 +d2Iij (tr) +dt2 +, +tr = t − r +c, +(2.2) +where G is the gravitational constant, c is the speed of light, and r is the distance from us +to the center of mass of the two galaxies or galaxy clusters. Moreover, Iij is the quadruple +moment +Iij(t) = +� +yiyjT 00(t, y)d3y = +� +yiyjρ(t, y)d3y, +(2.3) +where T µν is energy-momentum tensor, ρ is energy density, and yi is the spatial coordinate. +Since the goal of our calculation is to acquire an estimation of the order of the magnitude of +the resulting signal, we can consider these two DM halos as mass points, with mass Ma and +position y(a)(t) at time t. Hence, the density ρ can be written as +ρ(t, y) = +� +a +Maδ3(y − y(a)(t)), +(2.4) +while the quadruple moment Iij(t) becomes +Iij(t) = +� +yiyjρ(t, y)d3y = +� +a +Mayi +(a)(t)yj +(a)(t). +(2.5) +– 2 – + +Finally, since the relative speed of two galaxies or galaxy clusters is much smaller than the +speed of light, we can use Newtonian mechanics to handle their dynamics. +For simplicity we write the equations in the center-of-mass frame of these two mass +points. By definition, we have +MArA + MBrB = 0, +(2.6) +where MA, MB are the masses of the mass points A and B, with rA, rB their position vectors. +From Newtonian mechanics we have +¨rA = − +GMB +|rA − rB|2 +rA +|rA|, +(2.7) +which using (2.6) gives +¨rA = −µB +rA +|rA|3 , +(2.8) +where we have defined µB ≡ +GMB +(1+ MA +MB )2 . Additionally, we assume that the two points are ini- +tially at infinite distance, their relative speed is v∞ = vA∞ +vB∞, and the impact parameter +is b = bA + bB. From Newtonian mechanics we know that the trajectory of each point is a +hyperbola and the two points are moving in a plane (we set this plane as z = 0 plane, and +thus rA = (xA, yA, 0)), while the total energy of the system is positive. Additionally, the +mass center of these two DM halos will not follow a hyperbolic trajectory at all times, in +order to acquire a collision. In Fig. 1 we depict an illustrative representation of the initial +conditions of the collision. +Figure 1. +An illustrative representation of the initial conditions of the collision. The two galaxies +or galaxy clusters are considered as points with masses MA and MB, where bA and bB are the impact +parameters. +Let us start with the beginning of the collision, when the two DM halos start moving +towards each other. For point A we have +(xA + aAeA)2 − (yA)2 = a2 +A, +(2.9) +– 3 – + +Y +MB +UB8 +bB +O center of mass +X +L= 8 +b A +VA8 +MAwhere +aA = +µB +(vA∞)2 , +(2.10) +eA = +� +1 + vA4∞b2 +A +(µB)2 , +(2.11) +a = aA + aB. +(2.12) +We proceed by defining λA through +eA sinh(λA) − λA = vA∞t +aA +, +(2.13) +hence +r1 +A = xA = aA [eA − cosh(λA)] , +(2.14) +r2 +A = yA = aA +�� +e2 +A − 1 sinh(λA) +� +. +(2.15) +Note that t = 0 corresponds to the time when the two mass points have the shortest distance. +In order to obtain the GW amplitude hij, we proceed to the calculation of the quadrupole +moment Iij(t) and its second time derivative. We have +Iij = MAri +Arj +A + MBri +Brj +B, +(2.16) +d2Iij +dt2 += MA(¨ri +Arj +A + ri +A¨rj +A + 2 ˙ri +A ˙rj +A) + MB(¨ri +Brj +B + ri +B¨rj +B + 2 ˙ri +B ˙rj +B). +(2.17) +From (2.14), (2.15) we find +˙xA = − +aA sinh(λA) +� +a3 +A +µB [eA cosh(λA) − 1] +, +(2.18) +¨xA = +µB(cosh(λA) − e) +a2 +A[eA cosh(λA) − 1]3 , +(2.19) +˙yA = +aA +� +e2 +A − 1 cosh(λA) +� +a3 +A +µB [eA cosh(λA) − 1] +, +(2.20) +¨yA = − +� +e2 +A − 1µB sinh(λA) +a2 +A[eA cosh(λA) − 1]3 , +(2.21) +and thus inserting into (2.17) we extract all the second time derivatives of the quadrupole +– 4 – + +moment Iij(t), namely +d2I11 +dt2 += µBMA{7eA cosh(λA) + eA[cosh(3λA) − 4eA] − 4 cosh(2λA)} +2aA[eA cosh(λA) − 1]3 ++µAMB{7eB cosh(λB) + eB[cosh(3λB) − 4eB] − 4 cosh(2λB)} +2aB[eB cosh(λB) − 1]3 +, +(2.22) +d2I12 +dt2 += − +� +e2 +A − 1µBMA sinh(λA){eA[cosh(2λA) + 3] − 4 cosh(λA)} +aA[eA cosh(λA) − 1]3 +− +� +e2 +B − 1µBMA sinh(λB){eB[cosh(2λB) + 3] − 4 cosh(λB)} +aB[eB cosh(λB) − 1]3 +, +(2.23) +d2I22 +dt2 += +� +e2 +A − 1 +� +µBMA[3eA cosh(λA) + eA cosh(3λA) − 4 cosh(2λA)] +2aA[eA cosh(λA) − 1]3 ++ +� +e2 +B − 1 +� +µAMB[3eB cosh(λB) + eB cosh(3λB) − 4 cosh(2λB)] +2aB[eB cosh(λB) − 1]3 +. +(2.24) +Figure 2. +The dimensionless components of the gravitational wave signal arising from a single +event of the collision of two DM halos, i.e., the collision of two galaxies or clusters of galaxies. The +left panel shows the ¯h11 component, the middle panel the ¯h12 component and the right panel the ¯h22 +component. The time t = 0 corresponds to the shortest distance between the two DM halos, that is +the moment in which ¯h11 and ¯h22 reach their peaks and ¯h12 exhibits the largest variation. We have +imposed the typical values MA = MB = 109M⊙, vA∞ = vB∞ = 300km/s, bA = bB = 104ly, and we +have assumed that the distance from Earth is ∼ 109ly. Time t is measured in seconds. +We can now use (2.2) in order to obtain the GW signal in the time domain. As typical +values we set MA = MB = 109M⊙, namely the order of mass of a (dwarf) galaxy, where M⊙ +is the mass of the Sun, and we use vA∞ = vB∞ = 300km/s, bA = bB = 104ly, which are +the typical values for galaxy collisions. Moreover, we assume that the collision happens at a +distance of 109ly from the Earth, which is roughly the distance of the source of GW150914. +Hence, we can estimate the magnitude of the GW signal. In Fig. 2 we present the obtained +dimensionless GW signal ¯hij, as a function of time t. Since t = 0 corresponds to the time of +shortest distance, the change rate of ¯hij is fastest at this time, as expected. As we observe, +the variation of ¯hij is of the order of 5 × 10−22 during the collision. However, this variation +– 5 – + +七 +-1.0×1017 -5.0×1016 +5.0 ×1016 +1.0×1017 +-1. × 10-22 +-3. × 10-22 +-5. × 10-22h12 +2.×10 +七 +-1.0×1017 -5.0×1016 +5.0 × 1016 +6 1.0×1017 +-1. × 10-22 +-2. × 10-22h22 +1.2538 × 10-18 +1.2534 × 10-18 +t +-1.0×1017 -5.0×1016 +5.0×1016 +1.0 ×1017corresponds to a large time scale (about 1015s), which implies that a single signal of this kind +of GW is extremely hard to be detected. Additionally, we can see that the evolution of ¯h12 +is faster than that of ¯h11, ¯h22, which implies that ¯h12 will be dominant in relatively higher +frequency than that of ¯h11, ¯h22. +We proceed by taking the Fourier transformation of ¯hij, in order to investigate its +spectrum. In particular, we use +˜¯hij(ω) = +� t=+∞ +t=−∞ +dt eiωt ¯hij(t) , +(2.25) +where ω = 2πf, with f the frequency. In Fig. 3 we present the dependence of ˜¯hij(f) on f. +As we observe, all ˜¯hij(f) obey the power law in a very good approximation for a very wide +frequency range. Besides, as ˜¯h11, ˜¯h22 ∝ 1/f2, while ˜¯h12 ∝ 1/f, we can infer that ˜¯h11, ˜¯h22 +will be dominant in the low frequency band while ˜¯h12 will be dominant in relatively high +frequencies. +Figure 3. The spectrum of the gravitational waves as a function of the frequency. The upper left panel +shows the ˜¯h11(f) component, the upper right panel the ˜¯h12(f) component and the right panel the ˜¯h22(f) +component. The blue dots represent the exact results at the time of shortest distance, while the red solid +curves are power-law fits, specifically ˜¯h11(f) ≈ 1.7 × 10−53(Hz/f)2, ˜¯h12(f) ≈ 1.2 × 10−38(Hz/f)2, +˜¯h22(f) ≈ 2.0 × 10−53(Hz/f)2. +3 +Effect on the stochastic gravitational wave background +In this section, we calculate the contribution of the DM halos collisions to the stochastic +gravitational wave background. Specifically, we integrate the gravitational wave spectrum of +– 6 – + +10-23 +10~24 +10~25 +10~26 +10~27 +10-28 +5.×10-15 10-14 +5. × 10-1.4 10-13 +5.× 10-13h11(f)f/H +21. × 10-22 +5. × 10-23 +1.×10-23 +5.× 10-24 +1. × 10-24 +5. × 10-25 +5. × 10-15 10-14 +5. × 10-1.4 10-13 +5.×10-13h12(f)f/H +210~23 +10~24 +10~25 +10-26 +10~27 +10-28 +5.× 10-15 10-14 +5. × 10-1.4 10-13 +5.× 10-13h22(f)f/H +2a single collision event over the number density of GW sources. +In principle, in order to compare a theoretical model with observations, one uses both +the fractional energy density spectrum Ωgw(f), as well as the characteristic strain amplitude +hc(f) [56]. They are related to the energy spectrum of GWB through the expression +π +4 f2h2 +c(f) = ρcΩgw(f) = dρgw(f) +d ln f , +(3.1) +where f is the frequency of GW detected on Earth, and ρc ≡ 3c2H2 +0/8πG is the critical +energy density. The energy spectrum of the stochastic GWB, dρgw +d ln f , can be written as +dρgw(f) +d ln f += +� ∞ +0 +dz +1 +1 + z +� +dξ dn +dzdξ +dE(ξ)gw +d ln fr +���� +fr=f(1+z) +, +(3.2) +with z the redshift at the GW emission. Additionally, dE(ξ)gw +d ln(fr) is the energy spectrum of a +single GW event, which is calculated through the analysis of the previous subsection, and fr +is the GW frequency in the rest frame of GW sources, and thus fr = (1 + z)f. +We mention that we denote the parameters related to the number density of GW sources +collectively by ξ = {ξ1, . . . , ξm}, and therefore +dn +dξ1...dξmdzdξ1 . . . dξmdz ≡ +dn +dξdzdξdz is the +number density of sources in the redshift interval [z, z + dz] and with source parameters +in the interval [ξ, ξ + dξ]. Hence, in the simple single event of two DM halos collision of +the previous section we have ξ = {M, x, v∞, b}, where M = MA + MB, x = MA/MB, +v∞ = vA∞ + vB∞ and b = bA + bB. +Let us now calculate the full distribution function +dn +dzdξ = +dn +dzdMdxdv∞db. As we have +checked numerically, the variance of b, v∞ has a minor effect on the final result, not affecting +the order of magnitude. Hence, it is a good approximation to omit the change of b, v∞, and +consider that ξ = {M, x}. Hence, we have +dρgw(f) +d ln f += +� ∞ +0 +dz +1 +1 + z +� +dξ dn +dzdξ +dE(ξ)gw +d ln fr +���� +fr=f(1+z) +≈ +� 10 +0 +dz +1 +1 + z +� Mmax=1015M⊙ +Mmin=109M⊙ +dM +� xmax=105 +xmin=1 +dx +dn +dzdMdx +dE(ξ)gw +d ln, fr +����� +fr=f(1+z) +, (3.3) +where the varying range of M and x is taken from [89]. +In the following subsections we will separately calculate the energy spectrum of a single +GW event dE(ξ)gw +d ln fr , and the number density of GW sources +dn +dzdMdx. +3.1 +Energy spectrum of a single GW event +The energy density of a single GW event can be calculated from the (traceless) second time +derivative of the quadrupole moment, namely [90] +dE(ξ)gw +d ln fr +≈ fr +2G +5c5 (2πfr)2( ¨˜Qij(M, x; fr))( ¨˜Qij(M, x; fr)), +(3.4) +– 7 – + +where Qij is the traceless quadrupole moment and ¨˜Qij is the Fourier transformation of the +second time derivative of Qij, which is related to Iij via +Q11 = 2 +3I11 − 1 +3I22, +(3.5) +Q22 = −1 +3I11 + 2 +3I22, +(3.6) +Q33 = −1 +3I11 − 1 +3I22, +(3.7) +Q21 = Q12 = I12, +(3.8) +while all other Qij are equal to zero. Now, from Newtonian mechanics Iij can be written as +¨˜Iij(M, x; fr) = 4 +� +x +(1 + x)3 + +1/x +(1 + 1/x)3 +� � +M +2 × 1012M⊙ +�2 ¨˜IG +ij (fr), +(3.9) +where x is the mass ratio of the two masses, and IG +ij is defined as Iij(M = 2×1012M⊙, x = 1). +Therefore, from the calculation of Section 2, we can extract the values of ¨˜IG +ij (fr) as +¨˜IG +11(fr) = 2.86 × 1021 +�Hz +fr +�2 +kg m2s−1, +(3.10) +¨˜IG +22(fr) = 5.72 × 1020 +�Hz +fr +�2 +kg m2s−1, +(3.11) +¨˜IG +12(fr) = ¨˜IG +21(fr) = 1.29 × 1037 +�Hz +fr +� +kg m2s−1. +(3.12) +Hence, inserting the above into (3.4) gives us the energy density of a single GW event. +3.2 +Number density of GW sources +Let us now calculate the number density of GW sources (per redshift, total mass and mass +ratio interval) +dn +dzdMdx. This number density is equal to the DM matter halos mergers rate, +which can be calculated by combining the extended Press-Schechter (EPS) theory [91] and +numerical simulations [89]: +dn +dzdMdx = nhalo(M, z)dω +dz +� +1 +nhalo +dnmerger +dωdx +� +, +(3.13) +where nhalo(M, z) is the number density of dark matter halos (per redshift per mass inter- +val in the co-moving space), ω = ω(z) is a redshift-dependent function given below, and +( +1 +nhalo +dnmerger +dωdx +) is the merger rate (at some ω) for a pair of DM halos with fixed total mass +M and mass ratio x. In the following we handle these terms separately. +We start with the definition of ω(z) [91] +ω(z) = 1.69 +D(z), +(3.14) +where D(z) is the linear growth rate of matter density. D(z) can be written as +D(z) = +1 +g(z = 0) +� g(z) +1 + z +� +, +(3.15) +– 8 – + +where a good approximation of g(z) is +g(z) ≈ 5 +2Ωm(z) +� +Ω4/7 +m (z) − ΩΛ(z) + [1 + Ωm(z)/2] [1 + ΩΛ(z)/70] +�−1 +, +(3.16) +with ΩΛ(z), Ωm(z) the density parameters of dark energy and matter sectors given by +ΩΛ(z) = ΩΛ,0 +E2(z); +Ωm(z) = Ωm,0(1 + z)3 +E2(z) +, +(3.17) +where the normalized Hubble function E(z) ≡ H(z)/H0 reads as +E(z) ≈ +� +ΩΛ,0 + Ωm,0(1 + z)3�1/2 , +(3.18) +with the value of the Hubble function at present time given as [64] +H0 ≈ 67.3 km s−1Mpc−1, +(3.19) +and with the values ΩΛ,0, Ωm,0 at present time taken as [64] +ΩΛ,0 ≈ 0.685, +(3.20) +Ωm,0 ≈ 0.317. +(3.21) +Note that in the above we consider that the underlying cosmology is ΛCDM concordance +scenario, i.e., the dark energy sector is the cosmological constant. +We continue by using the EPS theory in order to write the formula of the number +density of DM halos nhalo. We consider that the halos merge when the redshift is between z +and z +dz , and that the emitted GW signals are detected at Earth at present. In co-moving +space those halos are in the volume ∆V = 4πr2(z)d(r(z)). Now, the EPS theory provides +the number density of DM halos nEPS(M, z) at some redshift z and mass M. Therefore, we +have +nhalo = 4πr2(z)dr(z) +dz nEPS(M, z), +(3.22) +where the radius in the co-moving space r(z) is [91] +r(z) = +c +H0 +� z +0 +dz +′ +1 +E(z +′), +(3.23) +while the formula of nEPS(M, z) is [91] +nEPS(M, z) = +� +2 +π +¯ρ +M2 +δc +σ exp +� +− δ2 +c +2σ2 +� ���� +d ln σ +d ln M +���� . +(3.24) +In the above expression ¯ρ = ρcΩm,0 is the mean density of the matter component, δc = ω = +1.69 +D(z), while σ(M) is the variance of the matter density perturbation which can be estimated +as [91] +σ(M) ≈ σ8 +� R +r8 +�−β +, +(3.25) +– 9 – + +with M = 4π +3 ¯ρR3 , σ8 ≈ 1, β ≈ 0.6 + 0.8(Ωm,0h), h = 0.673 , and r8 = 8 Mpc h−1, leading to +���� +d ln σ +d ln M +���� = β +3 . +(3.26) +Finally, the last term of (3.13), namely ( +1 +nhalo +dnmerger +dωdx +) (dimensionless since both ω, x +are dimensionless), can be found in [89] and it is given by +� +1 +nhalo +dnmerger +dωdx +� += A +� +M +1012M⊙ +�α +xb exp [(˜x/x)γ] , +(3.27) +where the best-fit parameters from simulations are A = 0.065, α = 0.15, b = −0.3, ˜x = 2.5, +γ = 0.5 [89]. +In summary, inserting (3.14), (3.22) and (3.27) into (3.13), provides the value of the +number density of GW sources +dn +dzdMdx. +3.3 +The energy spectrum of the stochastic gravitational wave background +We have now all the ingredients needed in order to calculate the energy spectrum of the +stochastic gravitational wave background. This is given by (3.3), in which the energy spec- +trum of a single GW event dE(ξ)gw +d ln fr +was calculated in subsection 3.1, while the number density +of GW sources +dn +dzdMdx was calculated in subsection 3.2. Assembling everything, we finally +obtain the stochastic gravitational wave background resulting from DM halos collisions in +the universe, which is calculated numerically and it is shown in Fig. 4. Additionally, for com- +parison we also depict the corresponding result of of the contribution of the super-massive +black-hole mergers (SMBHM), which follows hc(f) ∝ f−2/3, as well as the current observa- +tional sensitivity [92]. +10 +16 +10 +14 +10 +12 +10 +10 +10 +8 +10 +6 +10 +4 +10 +2 +Frequency f /Hz +10 +34 +10 +30 +10 +26 +10 +22 +10 +18 +10 +14 +10 +10 +Characteristic Strain hc +DM halos collisions +SMBHM +current observation sensitivity +Figure 4. The characteristic strain hc(f) as a function of the frequency of the stochastic gravita- +tional wave background created by DM halos, namely galaxies and galaxy clusters, collisions (blue-solid +curve). For comparison, with the green-dashed curve we depict the corresponding result of of the con- +tribution of the super-massive black-hole mergers (SMBHM), which follows hc(f) ∝ f −2/3. Finally, +with the red-dotted curve we present the current observational sensitivity [92]. +– 10 – + +As we can see, the contribution of GW radiated from the collisions of DM halos, namely +galaxies and galaxy clusters, is quite small comparing to other sources. In the pulsar timing +array (PTA) band, where f ≈ 10−9Hz, and where the current observational limit is hc ≈ +10−15 [93], we obtain an effect of the order of hc ≈ 10−30. Nevertheless, in very low frequency +band hc will be larger. In general, with current observational sensitivity the effect of the DM +halos collisions on the stochastic gravitational wave background cannot be detected, however +it will be accessible in the future, in which case one could use the advanced techniques of +distinguishing signal overlap in order to isolate it from other sources [94–97]. +Note that +one could try to extend the analysis, by considering, instead of point masses, a group of +mass points with Navarro, Frenk & White (NFW) density profile [91] to simulate DM halo +collisions, nevertheless the results are expected to be at the same order of magnitude. +4 +Conclusions +In this work we investigated for the first time the effect of the dark matter halos collisions, +namely collisions of galaxies and galaxy clusters, through gravitational bremsstrahlung, on +the stochastic gravitational wave background. +In order to achieve this goal, we first calculated the gravitational wave signal of a single +DM halo collision event. As an estimation of the order of magnitude, we handled the two +DM halos as mass points. Furthermore, since the strength of such GW signals is weak, we +adopted linear perturbation theory of General Relativity, namely we extracted the GW signal +using the second time derivative of the quadruple moment. Additionally, since the velocity +of DM halos is small, we applied non-relativistic Newtonian Mechanics. Hence, we extracted +the GW signal through bremsstrahlung from a single DM halo collision. As we showed, ¯hij is +of the order of 10−22, and it becomes maximum at the time of shortest distance as expected. +However, since such an event typically corresponds to durations of the order of 1015s, we +deduce that a single signal of this kind of GW is extremely hard to be detected. +As a next step we proceeded to the calculation of the energy spectrum of the collective +effect of all DM halos collisions in the Universe. This can arise by the energy spectrum of +a GW signal radiated by a single collision, multiplied by the DM halo collision rate, and +integrating over the whole Universe. +Firstly, knowing the signal of a single collision we +calculated its energy spectrum. Secondly, concerning the DM halo collision rate we showed +that it is given by the product of the number density of DM halos, which is calculated by the +EPS theory, with the collision rate of a single DM halo, which is given by simulation results, +with a function of the linear growth rate of matter density through cosmological evolution. +Hence, integrating over all mass and distance ranges, we finally extracted the spectrum of +the stochastic gravitational wave background created by DM halos collisions. +As we saw, the resulting contribution to the stochastic gravitational wave background +is of the order of hc ≈ 10−30 in the pulsar timing array (PTA) band of f ≈ 10−9Hz, much +smaller than other GW sources, such as super-massive black-hole mergers. However, in very +low frequency band, hc is larger. With current observational sensitivity it cannot be detected, +nevertheless it may be accessible by PTA in the future, where techniques of distinguishing +signal overlap should be used in order to isolate it and use it for cosmological studies. +In summary, with the current and future significant advance in gravitational-wave as- +tronomy, and in particular with the tremendous improvement on the sensitivity bounds that +Collaborations like Laser Interferometer Space Antenna (LISA), Einstein Telescope (ET), +Cosmic Explorer (CE), etc will bring, it is both interesting and necessary to investigate all +– 11 – + +possibles contributions to the stochastic gravitational wave background. And the gravita- +tional bremsstrahlung during galaxies and galaxy clusters collisions is one of them. +Acknowledgments +We are grateful to Yifu Cai, Jiewen Chen, Zihan Zhou and Jiarui Li for helpful discussions. +This work is supported in part by the National Key R&D Program of China (2021YFC2203100), +by the NSFC (11961131007, 11653002), by the Fundamental Research Funds for Central Uni- +versities, by the CSC Innovation Talent Funds, by the CAS project for young scientists in +basic research (YSBR-006), by the USTC Fellowship for International Cooperation, and by +the USTC Research Funds of the Double First-Class Initiative. ENS acknowledges partici- +pation in the COST Association Action CA18108 “Quantum Gravity Phenomenology in the +Multimessenger Approach (QG-MM)”. 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Woan, Rapid determination of LISA sensitivity to +extreme mass ratio inspirals with machine learning, 2212.06166. +– 17 – + diff --git a/CdE0T4oBgHgl3EQfgQEc/content/tmp_files/load_file.txt b/CdE0T4oBgHgl3EQfgQEc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d4e8eb2950ef8b9be6289528a785ee9906075f05 --- /dev/null +++ b/CdE0T4oBgHgl3EQfgQEc/content/tmp_files/load_file.txt @@ -0,0 +1,1077 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf,len=1076 +page_content='Prepared for submission to JCAP Stochastic gravitational wave background from the collisions of dark matter halos Qiming Yana,b Xin Rena,b Yaqi Zhaoa,b Emmanuel N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Saridakisc,a,d aDeep Space Exploration Laboratory/School of Physical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China bCAS Key Laboratory for Researches in Galaxies and Cosmology/Department of Astronomy, School of Astronomy and Space Science, University of Science and Technology of China, Hefei, Anhui 230026, China cNational Observatory of Athens, Lofos Nymfon, 11852 Athens, Greece dDepartamento de Matem´aticas, Universidad Cat´olica del Norte, Avda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Angamos 0610, Casilla 1280 Antofagasta, Chile E-mail: asadoubi233@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='cn, rx76@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='cn, zxmyg86400@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='cn, msaridak@noa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='gr Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' We investigate for the first time the effect of the dark matter (DM) halos colli- sions, namely collisions of galaxies and galaxy clusters, through gravitational bremsstrahlung, on the stochastic gravitational wave background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' We first calculate the gravitational wave signal of a single collision event, assuming point masses and linear perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Then we proceed to the calculation of the energy spectrum of the collective effect of all dark matter collisions in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Concerning the DM halo collision rate we show that it is given by the product of the number density of DM halos, which is calculated by the extended Press-Schechter (EPS) theory, with the collision rate of a single DM halo, which is given by simulation results, with a function of the linear growth rate of matter density through cos- mological evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Hence, integrating over all mass and distance ranges, we finally extract the spectrum of the stochastic gravitational wave background created by DM halos collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' As we show, the resulting contribution to the stochastic gravitational wave background is of the order of hc ≈ 10−30 in the pulsar timing array (PTA) band of f ≈ 10−9Hz, much smaller than other GW sources, such as super-massive black-hole mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' However, in very low frequency band, it is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' With current observational sensitivity it cannot be detected, nevertheless it may be accessible by PTA in the future, where techniques of distinguishing signal overlap should be used in order to isolate it and use it for cosmological studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='02414v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='CO] 6 Jan 2023 Contents 1 Introduction 1 2 Gravitational waves emitted during a single collision 2 3 Effect on the stochastic gravitational wave background 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='1 Energy spectrum of a single GW event 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='2 Number density of GW sources 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='3 The energy spectrum of the stochastic gravitational wave background 10 4 Conclusions 11 1 Introduction Recently, the gravitational wave (GW) detecting technology has been developing rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In 2015, the detection of binary black holes merger GW150914 by the LIGO experimental cooperation signaled the first detection of gravitational waves [1], while in 2017, the joint detection of GW170817 [2] and GRB170817A [3] opened the new era of multi-messenger astronomy [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In general, with the increasing amount of detected gravitational wave events [5] one has improved statistics that allows to track the history of the universe [6, 7] and impose bounds on various cosmological parameters [8, 9], as well as constrain various theories of gravity [10–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Moreover, for different frequencies and types of gravitational wave sources, various detection means have been designed and implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Besides ground-based laser interferometers such as LIGO, Virgo and KAGRA, which probe high frequency bands (10 − 104 Hz), space-based laser interferometers such as LISA [15, 16] for intermediate frequency gravitational waves (10−4 − 1 Hz), and the pulsar timing array (PTA) [17–20] for lower frequency bands (10−9 − 10−6 Hz), are also raised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' These observational avenues allow us to acquire rich information from GWs of different types and sources, among which stochastic gravitational wave background is attracting increasing interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Stochastic gravitational wave background (GWB) is a type of random background signal that exists in an analogous way to the cosmic microwave background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' The contribution of GWB can be roughly divided into cosmological sources and astrophysical sources [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Astrophysical originated GWB contains all types of unresolved GW emitting events, including binary black hole mergers [22–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' These signals can provide information about astrophysical source populations and processes over the history of the universe [29–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' On the other hand, cosmological originated GWB mainly involves primordial gravitational perturbations during the inflation epoch [33–35], or perturbations arising from primordial black holes fluctuations [36–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' GW signals typically remain unaffected during their propagation, and thus they can provide valuable information about the very early stages of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' For instance, different inflationary models can lead to different predictions for the GWB spectrum [40–52], and thus GWB can be used as a probe of this primordial universe epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Since GWB can provide us with important astrophysical and cosmological probes, it is crucial to understand its composition and properties [53–63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' On the other hand, according to observations, dark matter (DM) constitutes a significant fraction of the energy density of the universe [64–66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Its microphysical nature and possible – 1 – interactions remain unknown [67–70], nevertheless we do know unambiguously that DM interacts gravitationally [71, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Current theory predicts that the main part of DM is concentrated in dark halos, which coincide in position with galaxy or galaxy clusters [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' These galaxies and galaxy clusters, and thus dark halos too, are typically accelerating and merging through their mutual attraction [74–76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Such processes can in principle release GW signal through gravitational bremsstrahlung [77–87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In this work, we are interested in investigating for the first time the possible GW signals that could be emitted through bremsstrahlung during dark halo merger and collisions, and their contribution to the stochastic GWB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In particular, we will first consider a single event of two DM halos collision, and we will calculate the emitted GW signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Then, we will calculate the energy spectrum contribution to the stochastic GWB, taking the DM halo collision rate into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' The structure of the article is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In Section 2 we analyze the GW emitted during the collision of two galaxies or two galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In Section 3 we integrate over redshift and DM halos parameters to extract the contribution to stochastic GWB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Finally, in Section 4 we conclude and discuss our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' 2 Gravitational waves emitted during a single collision In this section, we aim at estimating the gravitational waves emitted during a single collision event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In particular, we calculate the GW radiated by the collision of two DM halos, which corresponds to the collision of two galaxies or two galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' According to observations, such a collision typically has a huge duration, which in turn implies that the energy radiated through GWs per unit time is not very large, and thus we can safely use linear perturbation theory in the involved calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Specifically, we use [88] gµν = ηµν + hµν, |hµν| ≪ 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='1) ¯hij(t, x) = 2G rc4 d2Iij (tr) dt2 , tr = t − r c, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='2) where G is the gravitational constant, c is the speed of light, and r is the distance from us to the center of mass of the two galaxies or galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Moreover, Iij is the quadruple moment Iij(t) = � yiyjT 00(t, y)d3y = � yiyjρ(t, y)d3y, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='3) where T µν is energy-momentum tensor, ρ is energy density, and yi is the spatial coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Since the goal of our calculation is to acquire an estimation of the order of the magnitude of the resulting signal, we can consider these two DM halos as mass points, with mass Ma and position y(a)(t) at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Hence, the density ρ can be written as ρ(t, y) = � a Maδ3(y − y(a)(t)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='4) while the quadruple moment Iij(t) becomes Iij(t) = � yiyjρ(t, y)d3y = � a Mayi (a)(t)yj (a)(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='5) – 2 – Finally, since the relative speed of two galaxies or galaxy clusters is much smaller than the speed of light, we can use Newtonian mechanics to handle their dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' For simplicity we write the equations in the center-of-mass frame of these two mass points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' By definition, we have MArA + MBrB = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='6) where MA, MB are the masses of the mass points A and B, with rA, rB their position vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' From Newtonian mechanics we have ¨rA = − GMB |rA − rB|2 rA |rA|, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='7) which using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='6) gives ¨rA = −µB rA |rA|3 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='8) where we have defined µB ≡ GMB (1+ MA MB )2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Additionally, we assume that the two points are ini- tially at infinite distance, their relative speed is v∞ = vA∞ +vB∞, and the impact parameter is b = bA + bB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' From Newtonian mechanics we know that the trajectory of each point is a hyperbola and the two points are moving in a plane (we set this plane as z = 0 plane, and thus rA = (xA, yA, 0)), while the total energy of the system is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Additionally, the mass center of these two DM halos will not follow a hyperbolic trajectory at all times, in order to acquire a collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' 1 we depict an illustrative representation of the initial conditions of the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' An illustrative representation of the initial conditions of the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' The two galaxies or galaxy clusters are considered as points with masses MA and MB, where bA and bB are the impact parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Let us start with the beginning of the collision, when the two DM halos start moving towards each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' For point A we have (xA + aAeA)2 − (yA)2 = a2 A, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='9) – 3 – Y MB UB8 bB O center of mass X L= 8 b A VA8 MAwhere aA = µB (vA∞)2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='10) eA = � 1 + vA4∞b2 A (µB)2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='11) a = aA + aB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='12) We proceed by defining λA through eA sinh(λA) − λA = vA∞t aA , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='13) hence r1 A = xA = aA [eA − cosh(λA)] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='14) r2 A = yA = aA �� e2 A − 1 sinh(λA) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='15) Note that t = 0 corresponds to the time when the two mass points have the shortest distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In order to obtain the GW amplitude hij, we proceed to the calculation of the quadrupole moment Iij(t) and its second time derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' We have Iij = MAri Arj A + MBri Brj B, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='16) d2Iij dt2 = MA(¨ri Arj A + ri A¨rj A + 2 ˙ri A ˙rj A) + MB(¨ri Brj B + ri B¨rj B + 2 ˙ri B ˙rj B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='17) From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='14), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='15) we find ˙xA = − aA sinh(λA) � a3 A µB [eA cosh(λA) − 1] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='18) ¨xA = µB(cosh(λA) − e) a2 A[eA cosh(λA) − 1]3 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='19) ˙yA = aA � e2 A − 1 cosh(λA) � a3 A µB [eA cosh(λA) − 1] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='20) ¨yA = − � e2 A − 1µB sinh(λA) a2 A[eA cosh(λA) − 1]3 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='21) and thus inserting into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='17) we extract all the second time derivatives of the quadrupole – 4 – moment Iij(t), namely d2I11 dt2 = µBMA{7eA cosh(λA) + eA[cosh(3λA) − 4eA] − 4 cosh(2λA)} 2aA[eA cosh(λA) − 1]3 +µAMB{7eB cosh(λB) + eB[cosh(3λB) − 4eB] − 4 cosh(2λB)} 2aB[eB cosh(λB) − 1]3 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='22) d2I12 dt2 = − � e2 A − 1µBMA sinh(λA){eA[cosh(2λA) + 3] − 4 cosh(λA)} aA[eA cosh(λA) − 1]3 − � e2 B − 1µBMA sinh(λB){eB[cosh(2λB) + 3] − 4 cosh(λB)} aB[eB cosh(λB) − 1]3 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='23) d2I22 dt2 = � e2 A − 1 � µBMA[3eA cosh(λA) + eA cosh(3λA) − 4 cosh(2λA)] 2aA[eA cosh(λA) − 1]3 + � e2 B − 1 � µAMB[3eB cosh(λB) + eB cosh(3λB) − 4 cosh(2λB)] 2aB[eB cosh(λB) − 1]3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='24) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' The dimensionless components of the gravitational wave signal arising from a single event of the collision of two DM halos, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=', the collision of two galaxies or clusters of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' The left panel shows the ¯h11 component, the middle panel the ¯h12 component and the right panel the ¯h22 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' The time t = 0 corresponds to the shortest distance between the two DM halos, that is the moment in which ¯h11 and ¯h22 reach their peaks and ¯h12 exhibits the largest variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' We have imposed the typical values MA = MB = 109M⊙, vA∞ = vB∞ = 300km/s, bA = bB = 104ly, and we have assumed that the distance from Earth is ∼ 109ly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Time t is measured in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' We can now use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='2) in order to obtain the GW signal in the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' As typical values we set MA = MB = 109M⊙, namely the order of mass of a (dwarf) galaxy, where M⊙ is the mass of the Sun, and we use vA∞ = vB∞ = 300km/s, bA = bB = 104ly, which are the typical values for galaxy collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Moreover, we assume that the collision happens at a distance of 109ly from the Earth, which is roughly the distance of the source of GW150914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Hence, we can estimate the magnitude of the GW signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' 2 we present the obtained dimensionless GW signal ¯hij, as a function of time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Since t = 0 corresponds to the time of shortest distance, the change rate of ¯hij is fastest at this time, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' As we observe, the variation of ¯hij is of the order of 5 × 10−22 during the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' However, this variation – 5 – 七 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='0×1017 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='0×1016 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='0 ×1016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='0×1017 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' × 10-22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' × 10-22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' × 10-22h12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='×10 七 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='0×1017 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='0×1016 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='0 × 1016 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='0×1017 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' × 10-22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' × 10-22h22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='2538 × 10-18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='2534 × 10-18 t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='0×1017 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='0×1016 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='0×1016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='0 ×1017corresponds to a large time scale (about 1015s), which implies that a single signal of this kind of GW is extremely hard to be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Additionally, we can see that the evolution of ¯h12 is faster than that of ¯h11, ¯h22, which implies that ¯h12 will be dominant in relatively higher frequency than that of ¯h11, ¯h22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' We proceed by taking the Fourier transformation of ¯hij, in order to investigate its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In particular, we use ˜¯hij(ω) = � t=+∞ t=−∞ dt eiωt ¯hij(t) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='25) where ω = 2πf, with f the frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' 3 we present the dependence of ˜¯hij(f) on f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' As we observe, all ˜¯hij(f) obey the power law in a very good approximation for a very wide frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Besides, as ˜¯h11, ˜¯h22 ∝ 1/f2, while ˜¯h12 ∝ 1/f, we can infer that ˜¯h11, ˜¯h22 will be dominant in the low frequency band while ˜¯h12 will be dominant in relatively high frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' The spectrum of the gravitational waves as a function of the frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' The upper left panel shows the ˜¯h11(f) component, the upper right panel the ˜¯h12(f) component and the right panel the ˜¯h22(f) component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' The blue dots represent the exact results at the time of shortest distance, while the red solid curves are power-law fits, specifically ˜¯h11(f) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='7 × 10−53(Hz/f)2, ˜¯h12(f) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='2 × 10−38(Hz/f)2, ˜¯h22(f) ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='0 × 10−53(Hz/f)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' 3 Effect on the stochastic gravitational wave background In this section, we calculate the contribution of the DM halos collisions to the stochastic gravitational wave background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Specifically, we integrate the gravitational wave spectrum of – 6 – 10-23 10~24 10~25 10~26 10~27 10-28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='×10-15 10-14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' × 10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='4 10-13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='× 10-13h11(f)f/H 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' × 10-22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' × 10-23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='×10-23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='× 10-24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' × 10-24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' × 10-25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' × 10-15 10-14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' × 10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='4 10-13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='×10-13h12(f)f/H 210~23 10~24 10~25 10-26 10~27 10-28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='× 10-15 10-14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' × 10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='4 10-13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='× 10-13h22(f)f/H 2a single collision event over the number density of GW sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In principle, in order to compare a theoretical model with observations, one uses both the fractional energy density spectrum Ωgw(f), as well as the characteristic strain amplitude hc(f) [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' They are related to the energy spectrum of GWB through the expression π 4 f2h2 c(f) = ρcΩgw(f) = dρgw(f) d ln f , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='1) where f is the frequency of GW detected on Earth, and ρc ≡ 3c2H2 0/8πG is the critical energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' The energy spectrum of the stochastic GWB, dρgw d ln f , can be written as dρgw(f) d ln f = � ∞ 0 dz 1 1 + z � dξ dn dzdξ dE(ξ)gw d ln fr ���� fr=f(1+z) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='2) with z the redshift at the GW emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Additionally, dE(ξ)gw d ln(fr) is the energy spectrum of a single GW event, which is calculated through the analysis of the previous subsection, and fr is the GW frequency in the rest frame of GW sources, and thus fr = (1 + z)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' We mention that we denote the parameters related to the number density of GW sources collectively by ξ = {ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' , ξm}, and therefore dn dξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='dξmdzdξ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' dξmdz ≡ dn dξdzdξdz is the number density of sources in the redshift interval [z, z + dz] and with source parameters in the interval [ξ, ξ + dξ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Hence, in the simple single event of two DM halos collision of the previous section we have ξ = {M, x, v∞, b}, where M = MA + MB, x = MA/MB, v∞ = vA∞ + vB∞ and b = bA + bB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Let us now calculate the full distribution function dn dzdξ = dn dzdMdxdv∞db.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' As we have checked numerically, the variance of b, v∞ has a minor effect on the final result, not affecting the order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Hence, it is a good approximation to omit the change of b, v∞, and consider that ξ = {M, x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Hence, we have dρgw(f) d ln f = � ∞ 0 dz 1 1 + z � dξ dn dzdξ dE(ξ)gw d ln fr ���� fr=f(1+z) ≈ � 10 0 dz 1 1 + z � Mmax=1015M⊙ Mmin=109M⊙ dM � xmax=105 xmin=1 dx dn dzdMdx dE(ξ)gw d ln, fr ����� fr=f(1+z) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='3) where the varying range of M and x is taken from [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In the following subsections we will separately calculate the energy spectrum of a single GW event dE(ξ)gw d ln fr , and the number density of GW sources dn dzdMdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='1 Energy spectrum of a single GW event The energy density of a single GW event can be calculated from the (traceless) second time derivative of the quadrupole moment, namely [90] dE(ξ)gw d ln fr ≈ fr 2G 5c5 (2πfr)2( ¨˜Qij(M, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' fr))( ¨˜Qij(M, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' fr)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='4) – 7 – where Qij is the traceless quadrupole moment and ¨˜Qij is the Fourier transformation of the second time derivative of Qij, which is related to Iij via Q11 = 2 3I11 − 1 3I22, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='5) Q22 = −1 3I11 + 2 3I22, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='6) Q33 = −1 3I11 − 1 3I22, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='7) Q21 = Q12 = I12, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='8) while all other Qij are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Now, from Newtonian mechanics Iij can be written as ¨˜Iij(M, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' fr) = 4 � x (1 + x)3 + 1/x (1 + 1/x)3 � � M 2 × 1012M⊙ �2 ¨˜IG ij (fr), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='9) where x is the mass ratio of the two masses, and IG ij is defined as Iij(M = 2×1012M⊙, x = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Therefore, from the calculation of Section 2, we can extract the values of ¨˜IG ij (fr) as ¨˜IG 11(fr) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='86 × 1021 �Hz fr �2 kg m2s−1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='10) ¨˜IG 22(fr) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='72 × 1020 �Hz fr �2 kg m2s−1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='11) ¨˜IG 12(fr) = ¨˜IG 21(fr) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='29 × 1037 �Hz fr � kg m2s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='12) Hence, inserting the above into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='4) gives us the energy density of a single GW event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='2 Number density of GW sources Let us now calculate the number density of GW sources (per redshift, total mass and mass ratio interval) dn dzdMdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' This number density is equal to the DM matter halos mergers rate, which can be calculated by combining the extended Press-Schechter (EPS) theory [91] and numerical simulations [89]: dn dzdMdx = nhalo(M, z)dω dz � 1 nhalo dnmerger dωdx � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='13) where nhalo(M, z) is the number density of dark matter halos (per redshift per mass inter- val in the co-moving space), ω = ω(z) is a redshift-dependent function given below, and ( 1 nhalo dnmerger dωdx ) is the merger rate (at some ω) for a pair of DM halos with fixed total mass M and mass ratio x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In the following we handle these terms separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' We start with the definition of ω(z) [91] ω(z) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='69 D(z), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='14) where D(z) is the linear growth rate of matter density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' D(z) can be written as D(z) = 1 g(z = 0) � g(z) 1 + z � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='15) – 8 – where a good approximation of g(z) is g(z) ≈ 5 2Ωm(z) � Ω4/7 m (z) − ΩΛ(z) + [1 + Ωm(z)/2] [1 + ΩΛ(z)/70] �−1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='16) with ΩΛ(z), Ωm(z) the density parameters of dark energy and matter sectors given by ΩΛ(z) = ΩΛ,0 E2(z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Ωm(z) = Ωm,0(1 + z)3 E2(z) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='17) where the normalized Hubble function E(z) ≡ H(z)/H0 reads as E(z) ≈ � ΩΛ,0 + Ωm,0(1 + z)3�1/2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='18) with the value of the Hubble function at present time given as [64] H0 ≈ 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='3 km s−1Mpc−1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='19) and with the values ΩΛ,0, Ωm,0 at present time taken as [64] ΩΛ,0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='685, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='20) Ωm,0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='21) Note that in the above we consider that the underlying cosmology is ΛCDM concordance scenario, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=', the dark energy sector is the cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' We continue by using the EPS theory in order to write the formula of the number density of DM halos nhalo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' We consider that the halos merge when the redshift is between z and z +dz , and that the emitted GW signals are detected at Earth at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In co-moving space those halos are in the volume ∆V = 4πr2(z)d(r(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Now, the EPS theory provides the number density of DM halos nEPS(M, z) at some redshift z and mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Therefore, we have nhalo = 4πr2(z)dr(z) dz nEPS(M, z), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='22) where the radius in the co-moving space r(z) is [91] r(z) = c H0 � z 0 dz ′ 1 E(z ′), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='23) while the formula of nEPS(M, z) is [91] nEPS(M, z) = � 2 π ¯ρ M2 δc σ exp � − δ2 c 2σ2 � ���� d ln σ d ln M ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='24) In the above expression ¯ρ = ρcΩm,0 is the mean density of the matter component, δc = ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='69 D(z), while σ(M) is the variance of the matter density perturbation which can be estimated as [91] σ(M) ≈ σ8 � R r8 �−β , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='25) – 9 – with M = 4π 3 ¯ρR3 , σ8 ≈ 1, β ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='6 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='8(Ωm,0h), h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='673 , and r8 = 8 Mpc h−1, leading to ���� d ln σ d ln M ���� = β 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='26) Finally, the last term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='13), namely ( 1 nhalo dnmerger dωdx ) (dimensionless since both ω, x are dimensionless), can be found in [89] and it is given by � 1 nhalo dnmerger dωdx � = A � M 1012M⊙ �α xb exp [(˜x/x)γ] , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='27) where the best-fit parameters from simulations are A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='065, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='15, b = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='3, ˜x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='5, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='5 [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In summary, inserting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='14), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='27) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='13), provides the value of the number density of GW sources dn dzdMdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='3 The energy spectrum of the stochastic gravitational wave background We have now all the ingredients needed in order to calculate the energy spectrum of the stochastic gravitational wave background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' This is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='3), in which the energy spec- trum of a single GW event dE(ξ)gw d ln fr was calculated in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='1, while the number density of GW sources dn dzdMdx was calculated in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Assembling everything, we finally obtain the stochastic gravitational wave background resulting from DM halos collisions in the universe, which is calculated numerically and it is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Additionally, for com- parison we also depict the corresponding result of of the contribution of the super-massive black-hole mergers (SMBHM), which follows hc(f) ∝ f−2/3, as well as the current observa- tional sensitivity [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' 10 16 10 14 10 12 10 10 10 8 10 6 10 4 10 2 Frequency f /Hz 10 34 10 30 10 26 10 22 10 18 10 14 10 10 Characteristic Strain hc DM halos collisions SMBHM current observation sensitivity Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' The characteristic strain hc(f) as a function of the frequency of the stochastic gravita- tional wave background created by DM halos, namely galaxies and galaxy clusters, collisions (blue-solid curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' For comparison, with the green-dashed curve we depict the corresponding result of of the con- tribution of the super-massive black-hole mergers (SMBHM), which follows hc(f) ∝ f −2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Finally, with the red-dotted curve we present the current observational sensitivity [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' – 10 – As we can see, the contribution of GW radiated from the collisions of DM halos, namely galaxies and galaxy clusters, is quite small comparing to other sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In the pulsar timing array (PTA) band, where f ≈ 10−9Hz, and where the current observational limit is hc ≈ 10−15 [93], we obtain an effect of the order of hc ≈ 10−30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Nevertheless, in very low frequency band hc will be larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In general, with current observational sensitivity the effect of the DM halos collisions on the stochastic gravitational wave background cannot be detected, however it will be accessible in the future, in which case one could use the advanced techniques of distinguishing signal overlap in order to isolate it from other sources [94–97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Note that one could try to extend the analysis, by considering, instead of point masses, a group of mass points with Navarro, Frenk & White (NFW) density profile [91] to simulate DM halo collisions, nevertheless the results are expected to be at the same order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' 4 Conclusions In this work we investigated for the first time the effect of the dark matter halos collisions, namely collisions of galaxies and galaxy clusters, through gravitational bremsstrahlung, on the stochastic gravitational wave background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In order to achieve this goal, we first calculated the gravitational wave signal of a single DM halo collision event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' As an estimation of the order of magnitude, we handled the two DM halos as mass points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Furthermore, since the strength of such GW signals is weak, we adopted linear perturbation theory of General Relativity, namely we extracted the GW signal using the second time derivative of the quadruple moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Additionally, since the velocity of DM halos is small, we applied non-relativistic Newtonian Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Hence, we extracted the GW signal through bremsstrahlung from a single DM halo collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' As we showed, ¯hij is of the order of 10−22, and it becomes maximum at the time of shortest distance as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' However, since such an event typically corresponds to durations of the order of 1015s, we deduce that a single signal of this kind of GW is extremely hard to be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' As a next step we proceeded to the calculation of the energy spectrum of the collective effect of all DM halos collisions in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' This can arise by the energy spectrum of a GW signal radiated by a single collision, multiplied by the DM halo collision rate, and integrating over the whole Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Firstly, knowing the signal of a single collision we calculated its energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Secondly, concerning the DM halo collision rate we showed that it is given by the product of the number density of DM halos, which is calculated by the EPS theory, with the collision rate of a single DM halo, which is given by simulation results, with a function of the linear growth rate of matter density through cosmological evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Hence, integrating over all mass and distance ranges, we finally extracted the spectrum of the stochastic gravitational wave background created by DM halos collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' As we saw, the resulting contribution to the stochastic gravitational wave background is of the order of hc ≈ 10−30 in the pulsar timing array (PTA) band of f ≈ 10−9Hz, much smaller than other GW sources, such as super-massive black-hole mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' However, in very low frequency band, hc is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' With current observational sensitivity it cannot be detected, nevertheless it may be accessible by PTA in the future, where techniques of distinguishing signal overlap should be used in order to isolate it and use it for cosmological studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' In summary, with the current and future significant advance in gravitational-wave as- tronomy, and in particular with the tremendous improvement on the sensitivity bounds that Collaborations like Laser Interferometer Space Antenna (LISA), Einstein Telescope (ET), Cosmic Explorer (CE), etc will bring, it is both interesting and necessary to investigate all – 11 – possibles contributions to the stochastic gravitational wave background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' And the gravita- tional bremsstrahlung during galaxies and galaxy clusters collisions is one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Acknowledgments We are grateful to Yifu Cai, Jiewen Chen, Zihan Zhou and Jiarui Li for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' This work is supported in part by the National Key R&D Program of China (2021YFC2203100), by the NSFC 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Berry and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' Woan, Rapid determination of LISA sensitivity to extreme mass ratio inspirals with machine learning, 2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content='06166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} +page_content=' – 17 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfgQEc/content/2301.02414v1.pdf'} diff --git a/D9E5T4oBgHgl3EQfUg90/content/tmp_files/2301.05544v1.pdf.txt b/D9E5T4oBgHgl3EQfUg90/content/tmp_files/2301.05544v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7888316fc2d2fbc00a08af2468d85a24624bff42 --- /dev/null +++ b/D9E5T4oBgHgl3EQfUg90/content/tmp_files/2301.05544v1.pdf.txt @@ -0,0 +1,497 @@ +UserSimCRS: A User Simulation Toolkit for Evaluating +Conversational Recommender Systems +Jafar Afzali +University of Stavanger +j.afzali@stud.uis.no +Aleksander Mark Drzewiecki +University of Stavanger +am.drzewiecki@stud.uis.no +Krisztian Balog +University of Stavanger +Stavanger, Norway +krisztian.balog@uis.no +Shuo Zhang +Bloomberg +London, United Kingdom +szhang611@bloomberg.net +ABSTRACT +We present an extensible user simulation toolkit to facilitate auto- +matic evaluation of conversational recommender systems. It builds +on an established agenda-based approach and extends it with sev- +eral novel elements, including user satisfaction prediction, persona +and context modeling, and conditional natural language generation. +We showcase the toolkit with a pre-existing movie recommender +system and demonstrate its ability to simulate dialogues that mimic +real conversations, while requiring only a handful of manually +annotated dialogues as training data. +CCS CONCEPTS +• Information systems → Recommender systems. +KEYWORDS +Conversational recommender systems; user simulation +ACM Reference Format: +Jafar Afzali, Aleksander Mark Drzewiecki, Krisztian Balog, and Shuo Zhang. +2023. UserSimCRS: A User Simulation Toolkit for Evaluating Conversational +Recommender Systems. In Proceedings of the Sixteenth ACM International +Conference on Web Search and Data Mining (WSDM ’23), February 27-March +3, 2023, Singapore, Singapore. ACM, New York, NY, USA, 4 pages. https: +//doi.org/10.1145/3539597.3573029 +1 +INTRODUCTION +Conversational recommender systems (CRSs) elicit user preferences +via multi-turn real-time interactions using natural language [6, 9]. +There has been a great deal of progress in recent years on various +aspects, including question-based user preference elicitation [5, +10, 29], multi-turn conversational recommendation strategies [12], +and natural language understanding and generation [13, 27]. A +major challenges that remains, however, is evaluation [6]. Due to +the dynamic nature of interactions, measuring performance on +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than ACM +must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, +to post on servers or to redistribute to lists, requires prior specific permission and/or a +fee. Request permissions from permissions@acm.org. +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +© 2023 Association for Computing Machinery. +ACM ISBN 978-1-4503-9407-9/23/02...$15.00 +https://doi.org/10.1145/3539597.3573029 +the conversation level is not possible using offline test collections. +While online evaluation with users of a live service is an option, +it is expensive and does not scale. A promising solution to these +issues is user simulation [1, 6]. The idea there is to build a simulated +user that mimics how a real human would respond in a given +dialogue situation [19, 25]. Simulation thus offers a repeatable and +reproducible means of evaluation. (We note that it is not meant to +replace, but rather to complement human evaluation.) +There is indeed an emerging focus in recent research on using +simulation for evaluating conversational information access sys- +tems in general [1, 3, 17, 20, 23] and conversational recommenders +in particular [25, 26]. The current work aims to contribute to the de- +velopment of novel CRSs by recognizing the need for better tooling +for user simulation. In particular, we provide an extensible open- +source toolkit that is designed specifically for evaluation. Our work +is unique in at least three regards. First, it focuses on the task of +conversational recommendation and hence place a strong emphasis +on both the recommendation-specific conversation flow and on the +human-likeness of the generated user utterances. Second, it centers +around evaluation as opposed to other uses of simulation (most +commonly, synthetic data generation for reinforcement learning). +Third, it is designed to work with existing CRSs, without needing ac- +cess to source code or knowledge of their inner workings. It merely +requires collecting and annotating a small sample of dialogues. +Building on an established agenda-based simulator [25], we intro- +duce novel components, motivated by recent research [17, 23, 26], +for modeling user satisfaction, persona and context, and condi- +tional natural language generation. Given its modular design, the +toolkit can also be easily extended with other modeling options or +additional components. The toolkit is comprised of two Python li- +braries, which are made publicly available on GitHub: DialogueKit1 +is a collection of generic and reusable dialogue components, and +UserSimCRS2 is an extensible user simulator built on top. +2 +RELATED WORK +While there are several efforts on simulation toolkits for recom- +mender systems [8, 11, 14, 16, 21], our work differs from those in +two major ways. First, we focus on the task of conversational recom- +mendations and hence place a strong emphasis on natural language +understanding and generation. Thus, unlike others that operate in +1https://github.com/iai-group/DialogueKit +2https://github.com/iai-group/UserSimCRS +arXiv:2301.05544v1 [cs.IR] 13 Jan 2023 + +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +Jafar Afzali, Aleksander Mark Drzewiecki, Krisztian Balog, & Shuo Zhang +Figure 1: Conceptual overview of the user simulator. The parts in blue follow [25], while the yellow ones are novel additions. +the “intent space,” we operate in the “language space.” Second, our +objective is system evaluation, as opposed to training end-to-end +recommender systems using reinforcement learning (RL). +Our toolkit implements an agenda-based simulator [18], building +on and extending the approach in [25]. Alternatively, model-based +simulation could also be employed as it has been done recently for +task-based dialogue systems. Shi et al. [22] demonstrate how to +build model-based user simulators that rely on a simple Seq2seq di- +alogue system with copy and attention mechanisms, to facilitate RL- +based dialogue system training. ConvLab-2 [28] is an open-source +toolkit that enables researchers to build task-oriented dialogue sys- +tems, where user simulators are provided to support end-to-end +evaluation. These simulators can be assembled by equipping a neu- +ral network-based user policy with NLU and NLG components. +Tseng et al. [24] propose a learning framework for developing dia- +logue systems that perform joint optimization with an LSTM-based +user simulator, which consists of a dialogue manager, an NLG model, +and a dialogue context encoder. The dialogue systems and user sim- +ulator models are pre-trained using supervised learning and then +fine-tuned using reinforcement learning based on the generated +dialogues. Importantly, such model-based approaches can also be +incorporated into our framework in the future. +3 +CONCEPTUAL OVERVIEW +The goal of user simulation is to mimic how real users would re- +spond in given dialogue situation [19, 25]. Conceptually, our user +simulator follows the architecture of a typical task-based dialogue +system, which consists of natural language understanding, response +generation, and natural language generation components. Addi- +tionally, there is a dedicated user modeling component; see Fig. 1. +We opt for a modular design, as opposed to an end-to-end trainable +system, in order to have complete control over how responses are +generated and to allow for flexible extensions. Our work builds on +and extends the approach proposed in [25] as detailed below. +Natural language understanding (NLU) is responsible for +obtaining a structured representation of text utterances. Conven- +tionally, it entails intent classification and entity recognition. Addi- +tionally, motivated by recent research [17, 23], we also include a +classifier for user satisfaction prediction.3 +Response generation is currently based on agenda-based simu- +lation [18], however, it could be replaced with other approaches in +the future. Following [25], response generation is based on an in- +teraction model, which is responsible for initializing the agenda and +3User satisfaction prediction is only used in the training stage to annotate dialogues. +updating it. Updates to the agenda can be summarized as follows: +if the agent responds in an expected manner, the interaction model +pulls the next action off the agenda; otherwise, it either repeats the +same action as the previous turn or samples a new action. +User modeling consists of three sub-components. The preference +model captures users’ likes and dislikes. Following [25], it is modeled +as a personal knowledge graph [2], where nodes correspond to items +and attributes. Novel to our work is the modeling of persona, which +can capture user-specific traits, e.g., patience or cooperativeness, +and context, which can characterize the situation of the user, e.g., +temporal (time of the day and weekday vs. weekend), relational +(alone vs. group setting), or conversational (user satisfaction). We +focus on contextual aspects as these represent a so far unexplored +area of user modeling [9] and there is evidence suggesting that +language usage depends on persona and context [15, 23]. +Natural language generation (NLG) is currently template- +based, that is, given the output of the response generation mod- +ule, a fitting textual response is chosen and may be instantiated +with preferences. Additionally, we extend the NLG such that it can +be conditioned on context. For example, user responses might be +shorter/longer depending on the time of the day or users could use +a stronger language when getting dissatisfied with the system. +4 +SOFTWARE ARCHITECTURE +The toolkit is written in Python and is based on a modular archi- +tecture to support additional components, different models, and +custom features to be added in the future. There are two main li- +braries that are stacked on each other: DialogueKit provides basic +dialogue management functionalities, while UserSimCRS contains +simulation-specific models and logic. See Fig. 2 for an overview of +the main packages and their dependencies. Both libraries are made +available in the Python Package Index (PyPI). +4.1 +DialogueKit +DialogueKit models dialogue participants (users and agents), do- +mains (which define the types of slots for a particular application), +utterances, and annotations as base concepts. Utterances may be +annotated with intents and slot-value pairs. DialogueKit currently +supports two models for annotation, a cosine classifier for intents +and a minimal pipeline DIET classifier [4] for slot-value pairs.4 A +dialogue connector is included to orchestrate and store the conver- +sation between participants (human-human, human-machine, or +4The DIET classifier can be used for intent detection as well. + +User Simulator +Natural Language +Response Generation +Understanding (NLU) +Agenda-based simulator +User satisfaction prediction +[Future simulators] +Interaction model +Conversational +Agent +金 +Natural Language +User Modeling +Generation (NLG) +<. +Context model +Conditional NLG +Preference model +PersonaUserSimCRS: A User Simulation Toolkit for Evaluating Conversational Recommender Systems +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +DialogueKit +UserSimCRS +Dialogue connector +Platforms +Utilities +Evaluator +NLU +Entity extractor +Intent classifier +Satisfaction classifier +NLG +Template-based NLG +Core components +Annotation +Intent +Dialogue +Utterance +Domain +… +Agenda-based +simulator +Interaction model +User modeling +Preference model +Context model +Persona +Items +Conditional NLG +Participant +Agent +User +Item +Ratings +Figure 2: Overview of the main packages (in yellow) with +some of the core modules highlighted (in white). Arrows in- +dicate intra-library dependencies (in blue) and inter-library +dependencies (in black). +machine-machine). Furthermore, the evaluation component pro- +vides functionality required to evaluate a set of conversations with +respect to standard metrics (such as AvgTurns and AvgSuccess). +4.2 +UserSimCRS +The UserSimCRS library implements the simulation-specific compo- +nents in Fig. 1, specifically, response generation and user modeling. +During a conversation, any time the user is asked to provide pref- +erences, the preference model is consulted. Context is modeled in +a generic way such that it can capture, among others, temporal, +relational, and conversational factors. The generation of user utter- +ances may be conditioned on the user’s context and persona. Next, +we elaborate on how to use UserSimCRS for system evaluation. +Note that the library may also be used for training agents, but that +is outside the focus of the current paper. +5 +SYSTEM EVALUATION USING SIMULATION +This section discusses how to employ simulation for evaluating an +existing CRS and illustrates this with a case study. +5.1 +Methodology +The main objective of simulation-based evaluation in this work +is to establish a relative comparison between two systems. These +may be different variants of the same CRS or two different systems. +Importantly, the user simulator needs to target the differences that +we care about. For the sake of illustration, assume that there is +a baseline conversational movie recommender that understands +movie genres and an improved version that also recognizes plot +keywords. Having a user simulator that asks only for genres but not +for plot keywords will not capture the differences between these +two systems. Therefore, as a general principle, the user simulator +needs to be co-developed with the CRS and customized to mimic +the targeted user behavior. +5.2 +Setting up Simulation +A unique feature of our toolkit is that it allows for the evaluation +of any existing CRS by treating it as a “black box.” That is, it does +not require access to the source code or assume knowledge of its +inner workings—it merely relies on observable behavior. Setting up +an existing CRS with our simulator involves the following steps: +(1) Prepare domain and item collection: A config file with domain- +specific slot names must be prepared for the preference model. +Additionally, a file containing the item collection is required. +(2) Provide preference data: Preference data is consumed in the +form of item ratings (user ID, item ID, and rating triples). +(3) Dialogue sample: A small sample of dialogues with the CRS +needs to be collected. The sample size depends on the complex- +ity of the system, in terms of action space and language variety, +but is generally in the order of 5-50 dialogues. +(4) Define interaction model: A config file containing the space +of user and agent intents (i.e., possible actions), as well as the set +of expected agent responses for each user intent, is required for +the interaction model. The baseline (CRSv1) interaction model +shipped with the UserSimCRS library offers a starting point, +which may be further tailored according to the behavior and +capabilities of the given CRS. +(5) Annotate sample: The sample of dialogues must contain utter- +ance-level annotations in terms of intents and entities, as this +is required to train the NLU and NLG components. Note that +the slots used for annotation should be the same as the ones +defined in the domain file (cf. Step 1) and intents should follow +the ones defined in the interaction model (cf. Step 4.). +(6) Define user model/population: Simulation is seeded with a +user population that needs to be characterized, for example, +in terms of the different contexts (e.g., weekday vs. weekend, +alone vs. group setting) and personas (e.g., patient and impa- +tient users). Further, the number of users to be generated is to +be specified. Each user will have their own preference model, +which may be instantiated by grounding it in actual preferences +(i.e., the ratings dataset given in Step 2). +(7) Train simulator: The NLU, NLG, and response generation +components of the simulator are trained using the annotated +dialogue sample. +(8) Run simulation: Running the simulator will generate a set of +simulated conversations for each user with the CRS and save +those to files. +(9) Perform evaluation: Evaluation takes the set of simulated +dialogues generated in the previous step as input, and measures +the performance of the CRS in terms of the metrics implemented +in DialogueKit. +5.3 +Case Study +To see our user simulator in action, we conducted a case study with +IAI MovieBot [7],5 which is an open-source conversational movie +recommender system. This required creating a connector agent in +DialogueKit, which can talk to IAI MovieBot via a RESTful API. +We followed the steps listed above to prepare the user simulator. +This included collecting a sample of 8 dialogues, configuring the +domain (with title, genre, and keyword as slots), and annotating +user and system utterances using intents (according to our CRSv1 +interaction model) and slot-value pairs. As it can be seen from the +sample dialogue in Fig. 3, the simulator could successfully complete +dialogues with the CRS, mimicking the behavior of users observed +in the training data it was exposed to. +5https://github.com/iai-group/MovieBot + +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +Jafar Afzali, Aleksander Mark Drzewiecki, Krisztian Balog, & Shuo Zhang +Figure 3: Sample dialogue between IAI MovieBot (Left, in +green) and the user simulator (Right, in blue). +6 +CONCLUSION AND FUTURE DIRECTIONS +We have presented a user simulation toolkit, organized into two +Python libraries around general dialogue management and specific +user simulation functionality, to facilitate research on both conver- +sational recommender systems and simulation-based evaluation. +The toolkit is shipped with solid baseline models for each of the +components, a detailed set of instructions, and a working exam- +ple with an existing CRS, in order to make it easy for researchers +and developers to start conducting simulation-based experiments. +Future work is concerned with extending the components with ad- +ditional modeling options, including alternatives to agenda-based +simulation. 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Please answer a few +DISCLOSE +more questions to help me find a good movie.Can you give +me a few keywords? For example, 'organized crime' or +'contact lens'", +Pixar animation +INQUIRE.ELICIT +Do you like: Cat City 8.4 96 min +DISCLOSE +Tell me something about it +REAVEAL +What would you like to know about Cat City? +INQUIRE.ITEMINFO +藍 +INQUIRE.MORE +Director name +Its directed by Béla Ternovszky. +INQUIRE.MORE +I like this recommendation +DISCLOSE.MORE +NOTE.ACCEPT +Please choose your next step. +I would like to quite now. +INQUIRE.NEXT +I hope you had a good experience. Bye. +COMPLETE +END \ No newline at end of file diff --git a/D9E5T4oBgHgl3EQfUg90/content/tmp_files/load_file.txt b/D9E5T4oBgHgl3EQfUg90/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5ad72817fe11c9c40fc86595bbc61cb8b62f6d4b --- /dev/null +++ b/D9E5T4oBgHgl3EQfUg90/content/tmp_files/load_file.txt @@ -0,0 +1,417 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf,len=416 +page_content='UserSimCRS: A User Simulation Toolkit for Evaluating Conversational Recommender Systems Jafar Afzali University of Stavanger j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='afzali@stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='uis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='no Aleksander Mark Drzewiecki University of Stavanger am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='drzewiecki@stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='uis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='no Krisztian Balog University of Stavanger Stavanger, Norway krisztian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='balog@uis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='no Shuo Zhang Bloomberg London, United Kingdom szhang611@bloomberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='net ABSTRACT We present an extensible user simulation toolkit to facilitate auto- matic evaluation of conversational recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' It builds on an established agenda-based approach and extends it with sev- eral novel elements, including user satisfaction prediction, persona and context modeling, and conditional natural language generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' We showcase the toolkit with a pre-existing movie recommender system and demonstrate its ability to simulate dialogues that mimic real conversations, while requiring only a handful of manually annotated dialogues as training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' CCS CONCEPTS Information systems → Recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' KEYWORDS Conversational recommender systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' user simulation ACM Reference Format: Jafar Afzali, Aleksander Mark Drzewiecki, Krisztian Balog, and Shuo Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' UserSimCRS: A User Simulation Toolkit for Evaluating Conversational Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' In Proceedings of the Sixteenth ACM International Conference on Web Search and Data Mining (WSDM ’23), February 27-March 3, 2023, Singapore, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' ACM, New York, NY, USA, 4 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='1145/3539597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='3573029 1 INTRODUCTION Conversational recommender systems (CRSs) elicit user preferences via multi-turn real-time interactions using natural language [6, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' There has been a great deal of progress in recent years on various aspects, including question-based user preference elicitation [5, 10, 29], multi-turn conversational recommendation strategies [12], and natural language understanding and generation [13, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' A major challenges that remains, however, is evaluation [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Due to the dynamic nature of interactions, measuring performance on Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' WSDM ’23, February 27-March 3, 2023, Singapore, Singapore © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' ACM ISBN 978-1-4503-9407-9/23/02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='1145/3539597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='3573029 the conversation level is not possible using offline test collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' While online evaluation with users of a live service is an option, it is expensive and does not scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' A promising solution to these issues is user simulation [1, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' The idea there is to build a simulated user that mimics how a real human would respond in a given dialogue situation [19, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Simulation thus offers a repeatable and reproducible means of evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' (We note that it is not meant to replace, but rather to complement human evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=') There is indeed an emerging focus in recent research on using simulation for evaluating conversational information access sys- tems in general [1, 3, 17, 20, 23] and conversational recommenders in particular [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' The current work aims to contribute to the de- velopment of novel CRSs by recognizing the need for better tooling for user simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' In particular, we provide an extensible open- source toolkit that is designed specifically for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Our work is unique in at least three regards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' First, it focuses on the task of conversational recommendation and hence place a strong emphasis on both the recommendation-specific conversation flow and on the human-likeness of the generated user utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Second, it centers around evaluation as opposed to other uses of simulation (most commonly, synthetic data generation for reinforcement learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Third, it is designed to work with existing CRSs, without needing ac- cess to source code or knowledge of their inner workings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' It merely requires collecting and annotating a small sample of dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Building on an established agenda-based simulator [25], we intro- duce novel components, motivated by recent research [17, 23, 26], for modeling user satisfaction, persona and context, and condi- tional natural language generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Given its modular design, the toolkit can also be easily extended with other modeling options or additional components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' The toolkit is comprised of two Python li- braries, which are made publicly available on GitHub: DialogueKit1 is a collection of generic and reusable dialogue components, and UserSimCRS2 is an extensible user simulator built on top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 2 RELATED WORK While there are several efforts on simulation toolkits for recom- mender systems [8, 11, 14, 16, 21], our work differs from those in two major ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' First, we focus on the task of conversational recom- mendations and hence place a strong emphasis on natural language understanding and generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Thus, unlike others that operate in 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='com/iai-group/DialogueKit 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='com/iai-group/UserSimCRS arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='05544v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='IR] 13 Jan 2023 WSDM ’23, February 27-March 3, 2023, Singapore, Singapore Jafar Afzali, Aleksander Mark Drzewiecki, Krisztian Balog, & Shuo Zhang Figure 1: Conceptual overview of the user simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' The parts in blue follow [25], while the yellow ones are novel additions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' the “intent space,” we operate in the “language space.” Second, our objective is system evaluation, as opposed to training end-to-end recommender systems using reinforcement learning (RL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Our toolkit implements an agenda-based simulator [18], building on and extending the approach in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Alternatively, model-based simulation could also be employed as it has been done recently for task-based dialogue systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' [22] demonstrate how to build model-based user simulators that rely on a simple Seq2seq di- alogue system with copy and attention mechanisms, to facilitate RL- based dialogue system training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' ConvLab-2 [28] is an open-source toolkit that enables researchers to build task-oriented dialogue sys- tems, where user simulators are provided to support end-to-end evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' These simulators can be assembled by equipping a neu- ral network-based user policy with NLU and NLG components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Tseng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' [24] propose a learning framework for developing dia- logue systems that perform joint optimization with an LSTM-based user simulator, which consists of a dialogue manager, an NLG model, and a dialogue context encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' The dialogue systems and user sim- ulator models are pre-trained using supervised learning and then fine-tuned using reinforcement learning based on the generated dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Importantly, such model-based approaches can also be incorporated into our framework in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 3 CONCEPTUAL OVERVIEW The goal of user simulation is to mimic how real users would re- spond in given dialogue situation [19, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Conceptually, our user simulator follows the architecture of a typical task-based dialogue system, which consists of natural language understanding, response generation, and natural language generation components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Addi- tionally, there is a dedicated user modeling component;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' We opt for a modular design, as opposed to an end-to-end trainable system, in order to have complete control over how responses are generated and to allow for flexible extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Our work builds on and extends the approach proposed in [25] as detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Natural language understanding (NLU) is responsible for obtaining a structured representation of text utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Conven- tionally, it entails intent classification and entity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Addi- tionally, motivated by recent research [17, 23], we also include a classifier for user satisfaction prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='3 Response generation is currently based on agenda-based simu- lation [18], however, it could be replaced with other approaches in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Following [25], response generation is based on an in- teraction model, which is responsible for initializing the agenda and 3User satisfaction prediction is only used in the training stage to annotate dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' updating it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Updates to the agenda can be summarized as follows: if the agent responds in an expected manner, the interaction model pulls the next action off the agenda;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' otherwise, it either repeats the same action as the previous turn or samples a new action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' User modeling consists of three sub-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' The preference model captures users’ likes and dislikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Following [25], it is modeled as a personal knowledge graph [2], where nodes correspond to items and attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Novel to our work is the modeling of persona, which can capture user-specific traits, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=', patience or cooperativeness, and context, which can characterize the situation of the user, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=', temporal (time of the day and weekday vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' weekend), relational (alone vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' group setting), or conversational (user satisfaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' We focus on contextual aspects as these represent a so far unexplored area of user modeling [9] and there is evidence suggesting that language usage depends on persona and context [15, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Natural language generation (NLG) is currently template- based, that is, given the output of the response generation mod- ule, a fitting textual response is chosen and may be instantiated with preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Additionally, we extend the NLG such that it can be conditioned on context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' For example, user responses might be shorter/longer depending on the time of the day or users could use a stronger language when getting dissatisfied with the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 4 SOFTWARE ARCHITECTURE The toolkit is written in Python and is based on a modular archi- tecture to support additional components, different models, and custom features to be added in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' There are two main li- braries that are stacked on each other: DialogueKit provides basic dialogue management functionalities, while UserSimCRS contains simulation-specific models and logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 2 for an overview of the main packages and their dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Both libraries are made available in the Python Package Index (PyPI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='1 DialogueKit DialogueKit models dialogue participants (users and agents), do- mains (which define the types of slots for a particular application), utterances, and annotations as base concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Utterances may be annotated with intents and slot-value pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' DialogueKit currently supports two models for annotation, a cosine classifier for intents and a minimal pipeline DIET classifier [4] for slot-value pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='4 A dialogue connector is included to orchestrate and store the conver- sation between participants (human-human, human-machine, or 4The DIET classifier can be used for intent detection as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' User Simulator Natural Language Response Generation Understanding (NLU) Agenda-based simulator User satisfaction prediction [Future simulators] Interaction model Conversational Agent 金 Natural Language User Modeling Generation (NLG) <.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Context model Conditional NLG Preference model PersonaUserSimCRS: A User Simulation Toolkit for Evaluating Conversational Recommender Systems WSDM ’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' February 27-March 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Singapore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Singapore ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='DialogueKit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='UserSimCRS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Dialogue connector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Platforms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Utilities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Evaluator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='NLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Entity extractor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Intent classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Satisfaction classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='NLG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Template-based NLG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Core components ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Annotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Intent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Dialogue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Utterance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Domain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Agenda-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='simulator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Interaction model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='User modeling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Preference model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Context model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Persona ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Items ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Conditional NLG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Participant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Agent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='User ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Item ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Ratings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Figure 2: Overview of the main packages (in yellow) with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='some of the core modules highlighted (in white).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Arrows in- dicate intra-library dependencies (in blue) and inter-library dependencies (in black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' machine-machine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Furthermore, the evaluation component pro- vides functionality required to evaluate a set of conversations with respect to standard metrics (such as AvgTurns and AvgSuccess).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='2 UserSimCRS The UserSimCRS library implements the simulation-specific compo- nents in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 1, specifically, response generation and user modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' During a conversation, any time the user is asked to provide pref- erences, the preference model is consulted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Context is modeled in a generic way such that it can capture, among others, temporal, relational, and conversational factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' The generation of user utter- ances may be conditioned on the user’s context and persona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Next, we elaborate on how to use UserSimCRS for system evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Note that the library may also be used for training agents, but that is outside the focus of the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 5 SYSTEM EVALUATION USING SIMULATION This section discusses how to employ simulation for evaluating an existing CRS and illustrates this with a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='1 Methodology The main objective of simulation-based evaluation in this work is to establish a relative comparison between two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' These may be different variants of the same CRS or two different systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Importantly, the user simulator needs to target the differences that we care about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' For the sake of illustration, assume that there is a baseline conversational movie recommender that understands movie genres and an improved version that also recognizes plot keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Having a user simulator that asks only for genres but not for plot keywords will not capture the differences between these two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Therefore, as a general principle, the user simulator needs to be co-developed with the CRS and customized to mimic the targeted user behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='2 Setting up Simulation A unique feature of our toolkit is that it allows for the evaluation of any existing CRS by treating it as a “black box.” That is, it does not require access to the source code or assume knowledge of its inner workings—it merely relies on observable behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Setting up an existing CRS with our simulator involves the following steps: (1) Prepare domain and item collection: A config file with domain- specific slot names must be prepared for the preference model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Additionally, a file containing the item collection is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' (2) Provide preference data: Preference data is consumed in the form of item ratings (user ID, item ID, and rating triples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' (3) Dialogue sample: A small sample of dialogues with the CRS needs to be collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' The sample size depends on the complex- ity of the system, in terms of action space and language variety, but is generally in the order of 5-50 dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' (4) Define interaction model: A config file containing the space of user and agent intents (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=', possible actions), as well as the set of expected agent responses for each user intent, is required for the interaction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' The baseline (CRSv1) interaction model shipped with the UserSimCRS library offers a starting point, which may be further tailored according to the behavior and capabilities of the given CRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' (5) Annotate sample: The sample of dialogues must contain utter- ance-level annotations in terms of intents and entities, as this is required to train the NLU and NLG components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Note that the slots used for annotation should be the same as the ones defined in the domain file (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Step 1) and intents should follow the ones defined in the interaction model (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' (6) Define user model/population: Simulation is seeded with a user population that needs to be characterized, for example, in terms of the different contexts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=', weekday vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' weekend, alone vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' group setting) and personas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=', patient and impa- tient users).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Further, the number of users to be generated is to be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Each user will have their own preference model, which may be instantiated by grounding it in actual preferences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=', the ratings dataset given in Step 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' (7) Train simulator: The NLU, NLG, and response generation components of the simulator are trained using the annotated dialogue sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' (8) Run simulation: Running the simulator will generate a set of simulated conversations for each user with the CRS and save those to files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' (9) Perform evaluation: Evaluation takes the set of simulated dialogues generated in the previous step as input, and measures the performance of the CRS in terms of the metrics implemented in DialogueKit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='3 Case Study To see our user simulator in action, we conducted a case study with IAI MovieBot [7],5 which is an open-source conversational movie recommender system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' This required creating a connector agent in DialogueKit, which can talk to IAI MovieBot via a RESTful API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' We followed the steps listed above to prepare the user simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' This included collecting a sample of 8 dialogues, configuring the domain (with title, genre, and keyword as slots), and annotating user and system utterances using intents (according to our CRSv1 interaction model) and slot-value pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' As it can be seen from the sample dialogue in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 3, the simulator could successfully complete dialogues with the CRS, mimicking the behavior of users observed in the training data it was exposed to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='com/iai-group/MovieBot WSDM ’23, February 27-March 3, 2023, Singapore, Singapore Jafar Afzali, Aleksander Mark Drzewiecki, Krisztian Balog, & Shuo Zhang Figure 3: Sample dialogue between IAI MovieBot (Left, in green) and the user simulator (Right, in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 6 CONCLUSION AND FUTURE DIRECTIONS We have presented a user simulation toolkit, organized into two Python libraries around general dialogue management and specific user simulation functionality, to facilitate research on both conver- sational recommender systems and simulation-based evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' The toolkit is shipped with solid baseline models for each of the components, a detailed set of instructions, and a working exam- ple with an existing CRS, in order to make it easy for researchers and developers to start conducting simulation-based experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Future work is concerned with extending the components with ad- ditional modeling options, including alternatives to agenda-based simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' We also plan to evaluate additional existing CRSs to ensure that our framework generalizes to diverse systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Hum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Behav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 121 (2021), 106795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' [16] David Rohde, Stephen Bonner, Travis Dunlop, Flavian Vasile, and Alexan- dros Karatzoglou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' RecoGym: A Reinforcement Learning Environ- ment for the problem of Product Recommendation in Online Advertising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' arXiv:1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='00720 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='IR] [17] Alexandre Salle, Shervin Malmasi, Oleg Rokhlenko, and Eugene Agichtein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Agenda-Based User Simulation for Bootstrapping a POMDP Dialogue System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' of NAACL ’07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 149–152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' [19] Jost Schatzmann, Karl Weilhammer, Matt Stuttle, and Steve Young.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' A Survey of Statistical User Simulation Techniques for Reinforcement-Learning of Dialogue Management Strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Knowl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 21, 2 (June 2006), 97–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' [20] Ivan Sekulić, Mohammad Aliannejadi, and Fabio Crestani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Evaluating Mixed-Initiative Conversational Search Systems via User Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' of WSDM ’22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 888–896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' [21] Bichen Shi, Makbule Gulcin Ozsoy, Neil Hurley, Barry Smyth, Elias Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Tragos, James Geraci, and Aonghus Lawlor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' PyRecGym: A Reinforcement Learning Gym for Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' of RecSys ’19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 491–495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' [22] Weiyan Shi, Kun Qian, Xuewei Wang, and Zhou Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' How to Build User Simulators to Train RL-based Dialog Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' of EMNLP-IJCNLP ’19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 1990–2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' [23] Weiwei Sun, Shuo Zhang, Krisztian Balog, Zhaochun Ren, Pengjie Ren, Zhumin Chen, and Maarten de Rijke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Simulating User Satisfaction for the Evaluation of Task-Oriented Dialogue Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' of SIGIR ’21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 2499–2506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' [24] Bo-Hsiang Tseng, Yinpei Dai, Florian Kreyssig, and Bill Byrne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Transferable Dialogue Systems and User Simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' of ACL ’21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 152–166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' [25] Shuo Zhang and Krisztian Balog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Evaluating Conversational Recommender Systems via User Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' of KDD ’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 1512–1520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' [26] Shuo Zhang, Mu-Chun Wang, and Krisztian Balog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Analyzing and Simulat- ing User Utterance Reformulation in Conversational Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' of SIGIR ’22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 133–143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' [27] Yongfeng Zhang, Xu Chen, Qingyao Ai, Liu Yang, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Bruce Croft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Towards Conversational Search and Recommendation: System Ask, User Respond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' of CIKM ’18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 177–186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' [28] Qi Zhu, Zheng Zhang, Yan Fang, Xiang Li, Ryuichi Takanobu, Jinchao Li, Baolin Peng, Jianfeng Gao, Xiaoyan Zhu, and Minlie Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' ConvLab-2: An Open-Source Toolkit for Building, Evaluating, and Diagnosing Dialogue Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' of ACL ’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 142–149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' [29] Jie Zou, Yifan Chen, and Evangelos Kanoulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Towards Question-Based Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' of SIGIR ’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' 881–890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Hi there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' I am IAl MovieBot, your movie recommending buddy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' I can recommend you movies based on your preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' I will ask you a few questions and based on your answers, I will try to find a movie for you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' ", Hello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' DISCLOSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='NON-DISCLOSE Do you have any specific genres in mind?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' DISCLOSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='NON-DISCLOSE How about Crime?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' INQUIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='ELICIT There are almost 5300 crime films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Please answer a few DISCLOSE more questions to help me find a good movie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='Can you give me a few keywords?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' For example, \'organized crime\' or \'contact lens\'", Pixar animation INQUIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='ELICIT Do you like: Cat City 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='4 96 min DISCLOSE Tell me something about it REAVEAL What would you like to know about Cat City?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' INQUIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='ITEMINFO 藍 INQUIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='MORE Director name Its directed by Béla Ternovszky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' INQUIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='MORE I like this recommendation DISCLOSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='MORE NOTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='ACCEPT Please choose your next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' I would like to quite now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' INQUIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content='NEXT I hope you had a good experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' Bye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} +page_content=' COMPLETE END' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'} diff --git a/D9FRT4oBgHgl3EQfAjeU/vector_store/index.faiss b/D9FRT4oBgHgl3EQfAjeU/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..34b17f562310dff255cbf85a3f1b6a59549703c8 --- /dev/null +++ b/D9FRT4oBgHgl3EQfAjeU/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b53e633e3ba4853f4219e62c68dbc6eef752c65dc371c9c05691b572b42c57ea +size 6094893 diff --git a/DNE5T4oBgHgl3EQfUA_Z/vector_store/index.pkl b/DNE5T4oBgHgl3EQfUA_Z/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..375a4372f23cbd678624bd6f07d4aaee66c7e71a --- /dev/null +++ b/DNE5T4oBgHgl3EQfUA_Z/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e324b0e93470070f31350495036c3b97dded9b821d6c049c173f92b92e63260f +size 132345 diff --git a/DdE2T4oBgHgl3EQfSQc5/content/tmp_files/2301.03790v1.pdf.txt b/DdE2T4oBgHgl3EQfSQc5/content/tmp_files/2301.03790v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a9488e212d9e3d724ee75c36f9dc4ec813fbf010 --- /dev/null +++ b/DdE2T4oBgHgl3EQfSQc5/content/tmp_files/2301.03790v1.pdf.txt @@ -0,0 +1,994 @@ +A Practical Runtime Security Policy +Transformation Framework for Software Defined +Networks +Yunfei Menga, Changbo Keb, Zhiqiu Huangc, Guohua Shenc, ChunQiang Liua, +Xiaojie Fenga +aCollege of Information Engineering, Qingdao Binhai University, Qingdao 266555, China +bSchool of Computer Science and Technology, Nanjing University of Posts and +Telecommunications, Nanjing 210023, China +cCollege of Computer Science and Technology, Nanjing University of Aeronautics and +Astronautics, Nanjing 211106, China +Abstract +Software-defined networking (SDN) has been widely utilized to enforce the se- +curity of traditional networks, thereby promoting the process of transforming +traditional networks into SDN networks. However, SDN-based security enforce- +ment mechanisms rely heavily on the security policies containing the underlying +information of data plane, such as MAC address, IP address or switch ports. +These security policies need to be specifically developed by the network opera- +tors, and loaded into the control plane by manual inputting. With increasing the +scale of underlying network, the current security policy management mechanism +will confront more and more challenges. The security policy transformation for +SDN networks is to research how to transform the high-level security policy +without containing the underlying information of data plane into the practical +flow entries used by the OpenFlow switches automatically, thereby implement- +ing the automation of security policy management. Based on this insight, a +practical runtime security policy transformation framework is proposed in this +paper. First of all, we specify the security policies used by SDN networks as a +system model of security policy (SPM). From the theoretical level, we establish +the system model for SDN network and propose a formal method to transform +SPM into the system model of flow entries automatically. From the practical +level, we propose a runtime security policy transformation framework to solve +the problem of how to find a connected path for each relationship of SPM in +the data plane, as well as how to generate the practical flow entries according +to the system model of flow entries. In order to validate the feasibility and ef- +fectiveness of the framework, we set up an experimental system and implement +the framework with POX controller and Mininet emulator. The experimental +results illustrate the framework can synchronously perceive the changes caused +by cutting down one edge or changing SPM, and keep the data plane holding +the security properties defined by SPM continuously at runtime. +Keywords: +SDN, security policy, model transformation, data plane. +Preprint submitted to Elsevier +January 11, 2023 +arXiv:2301.03790v1 [cs.CR] 10 Jan 2023 + +1. Introduction +Software-defined networking (SDN) is a novel networking technique or ar- +chitecture that changes the limitation of traditional network infrastructures by +breaking the vertical integration, decoupling the control logics from the underly- +ing forwarding devices, promoting the centralization of control and introducing +the abilities to program the network directly[1]. In SDN networks, the control +logics of network, such as routing, traffic engineering or security policy devel- +oped in the application plane, are loaded into the control plane via the north- +bound interfaces (NBI) and transformed into a set of forward entries used by +the OpenFlow switches. After that, the control plane distributes the generated +flow entries to the associated switches in the data plane via the southbound +interfaces (SBI). Because of its programmable, centralized intelligent control +as well as global traffic view, SDN has been widely utilized to enhance the +security of tradition networks, thereby promoting the process of transforming +traditional networks into SDN networks. For instances, Garay et al.[2] proposed +a SDN-based network access control mechanism, flownac, which is a centralized +EAP (extensible authentication protocol) for IEEE 802.1x wireless local area +network (WLAN). Yakasai et al.[3] proposed a network access control mech- +anism, flowidentity. This mechanism integrates EAP security authentication +mechanism into the SDN controller. Hu et al.[4] proposed a dynamic firewall +mechanism, flowguard, based on SDN. Koerner et al.[5] proposed a device se- +curity authentication mechanism based on MAC address and SDN. +However, SDN-based security enforcement mechanisms rely heavily on the +security policies containing the underlying information of the data plane, such +as MAC address, IP address or switch ports. These security policies need to +be specifically developed by the network operators, and loaded into the control +plane by means of the manual inputting. With increasing the scale of underlying +network, the current security policy management mechanism will confront more +and more challenges. First of all, it is nearly impossible for any operator to +completely understand all the information of underlying network. In addition, +with the emergence of multi-controller SDN[6], network operators need to man- +age a variety of heterogeneous controllers at the same time. In this case, the +same security policy often needs to be developed and deployed for the different +types of controller, which inevitably increases the complexity and difficulty for +network management. Therefore, a novel security policy management mecha- +nism which can be completely transparent to the underlying information of data +plane is urgently needed for SDN networks. That is, it can permit the operators +only to define the high-level security policy without containing any underlying +information, then by means of the security policy transformation, the high-level +security policy can be automatically transformed into its corresponding flow +entries used by the OpenFlow switches in the data plane, thereby implementing +the automation of security policy management for SDN networks. +2 + +Based on these insights, we have proposed a security policy model transfor- +mation and verification approach for SDN networks and published the approach +in our previous paper[7]. In that paper, we proposed a security policy transfor- +mation method to transform the high-level security policy model (SPM) without +containing the underlying information into its corresponding low-level security +policy model (LSPM) containing the underlying information. +To verify the +soundness of proposed security policy model transformation method, we further +proposed a security policy verification method and proved that the problem of +whether the data plane can satisfy the security properties defined by SPM is +equivalent to the problem of searching the connected paths related with SPM in +the data plane, that is, as long as each access control relationship Ri ∈ SPM can +be transformed into a corresponding connected path Pi in the data plane, next +transforms Pi into a set of flow entries used by the OpenFlow switches, then +the data plane must can hold the security properties defined by SPM. However, +that paper only proposed the method from the theoretical level, and did not +specifically implement this method. Moreover, it did not solve the problem of +how to find a connected path for each relationship of SPM in the data plane, +and how to transform LSPM into the practical flow entries used by the switches. +And based on the theoretical foundation of that paper, we propose a run- +time security policy transformation framework for SDN networks in this paper. +First of all, this paper further improves the system model of SDN networks and +solves the problem of how to transform SPM into the flow entries used by the +OpenFlow switches from the theoretical level. Moreover, this paper proposes +a runtime security policy transformation framework from the practical level, +thereby solving the problem of how to find a connected path Pi for each rela- +tionship Ri ∈ SPM in the data plane, as well as how to transform the system +model of flow entries into the practical flow entries used by the switches at run- +time. In addition, this paper further implements the proposed framework with +an experimental system. The experimental result illustrate the framework is +completely effective at runtime. +Hence, the contributions of this paper can be summarized as follows: +• We specify the security policies used by SDN networks, such as access +control policies or firewall policies, as a system model of security policy (SPM). +SPM is of a high-level system model without containing any underlying infor- +mation of data plane. +• From the theoretical level, we establish the system model for SDN network, +and propose a formal method to transform SPM into the system model of flow +entries automatically. The system model of flow entry is of a low-level system +model containing the underlying information of data plane. +• From the practical level, we propose a runtime security policy transforma- +tion framework which consists of the security policy module, topology discovery +module, runtime monitoring module, path generation module and flow entry +generation module. Leveraging these functional modules, the framework can +solve the problem of how to find a connected path for each relationship of SPM +in the data plane, how to transform the path into the system model of flow en- +tries, as well we how to generate the practical flow entries by using the system +3 + +model of flow entries. +• In order to validate the feasibility and effectiveness of the framework, we +set up an experimental system and implement the framework by using POX con- +troller and Mininet emulator. The experimental result illustrate the framework +is completely effective at runtime. +The remainder of this paper is structured as follows. Section 2 discusses +some related works. Section 3 proposes the system model and elaborates on +how to transform SPM into the flow entries from the theoretical level. Sec- +tion 4 proposes the runtime security policy transformation framework from the +practical level and introduces its functional modules. Section 5 implements the +framework with an experimental system and elaborates on how to evaluate the +effectiveness and performance of the framework. Finally, Section 6 concludes +this paper and presents some future directions. +2. Related Work +In this section, we discuss some research works related with the policy model +transformation and the security policy verification. +2.1. Policy Model Transformation +According to the definitions of model-driven architecture (MDA), the model +transformation refers to the process of transforming the platform independent +model (PIM) to its corresponding platform specific model (PSM)[8]. As far as +the literatures we have read, the researches towards the policy model transforma- +tion can be roughly divided into three categories, they are the template-based +transformation, RBAC-oriented transformation as well as the transformation +based on the system model and mapping rules[9]. Due to the limitation of tem- +plate, the template-based model transformation has very limited transformation +capability. Generally, RBAC-oriented model transformation[10] is only suitable +for transforming RBAC (role-based access control) policies, and does not have +enough capability to describe the complex system, so that these two methods +are not suitable for SDN networks. +At present, the model transformation based on the system model and map- +ping rules has been widely used for transforming the policy models. The main +idea of this method can be summarized as follows: (1) System Model: it de- +fines the objects of system and the relationship between the system objects; +(2) Policy Model: it defines the policy object and the relationship between the +policy objects; (3) Mapping Rules: it establishes the mapping rules between the +upper-level policy objects and the lower-level system objects[11][12]. The trans- +formation based on the system model and mapping rules first establishes the +policy model and the system model which can describe the underlying system, +then establishes the mapping rules between the policy objects and the system +objects, then transforms the upper-level policy model into its corresponding +lower-level policy model by means of the established mapping rules. In partic- +ular, Davy et al.[13] proposed a policy model transformation method based on +4 + +mapping rules, in which the policy model is defined as a tuple ( event, condi- +tion, behavior, subject, object ) and used the ontology to establish the mapping +rules between the different system layers. Luck et al.[14] proposed a method +to transform RBAC model defined in service layer into the policy model used +in the system layer. In this method, the system model is divided into three +layers: roles and object (RO), subject and resources (SR) and processes and +hosts (PH), and the mapping rules between the three layers have been estab- +lished. Based on the Luck’s research, Porto et al.[15] further decomposes the +PH layer into two sub layers, namely DAS (diagram abstract subsystem) layer +and PH layer. DAS layer is mainly used to describe the network topology in +the original PH layer, while PH layer is used to describe the specific network +information in DAS layer. In addition, the authors also proposed a policy ver- +ification framework, which can be used to verify the consistency problems in +the process of policy transformation. In addition, Lampson et al.[16] proposed +a network policy model transformation method for the distributed computing +environment. Maullo et al.[17] proposed a policy transformation system based +on the first-order predicate logics, which transforms the high-level policy model +into the low-level network configuration policy through the network topology +and other information. Nanxi et al.[18] proposed a SDN-oriented access control +policy transformation framework. In this paper, In this paper, we also propose a +security policy transformation framework based on the system model and map- +ping rules. We first establish the system model of security policy (SPM) and +data plane, then establish the transformation rules between the policy objects +of SPM and the objects of the data plane, thereby transforming SPM into the +system model of flow entries automatically. +2.2. Security Policy Verification +To assure the information systems running securely, security mechanisms +of information system need to be validated whether it can satisfy the security +properties defined by the security policy. The traditional validation methods +based on the testing and simulation can only confirm the system can work prop- +erly under the different testing scenarios, but it is difficult to find some hidden +scenarios that occur with little probability. Formal verification methods have +been applied to overcome the shortcomings existed in the traditional valida- +tion methods. At present, the formal verification methods for validating the +security policy mainly include theorem proving and model checking[19]. The- +orem proving is unsuitable to validate the properties of complex systems due +to its lower efficiency. Model checking[20] can be used to validate whether the +system model can satisfy the expected dynamic behaviors and specific static +properties. +Model checking technique has been widely used for the security +policy verification. For instances, Al-Shaer et al.[21] proposed a static policy +inconsistency detection method for the firewall policies of network. Bandara et +al.[22] proposed a security policy verification framework based on event calcu- +lus (EC) and used the reasoning techniques for the policy conflict identification. +May et al.[23] verified the privacy policies by means of an asynchronous model +checker. Rubio-Loyola et al.[24] proposed a goal-oriented policy refinement and +5 + +conflict detection method by means of the model checking technique and linear +temporal Logic (LTL). Graham et al.[25] proposed a policy conflict detection +method with the model checking and an extended decision table. Baliosan and +Serrat[26] proposed a specific finite automata based method for the policy con- +flict detection. +3. Problem Formalization +The security policy transformation for SDN networks is to research how to +transform the high-level security policy without containing the underlying infor- +mation into the set of practical flow entries used by the OpenFlow switches in +the data plane automatically, thereby implementing the automation of security +policy management in SDN network. In the following of this section, we first +establish the system model for SDN network, then propose a formal method to +transform the security policy (SPM) into the system model of flow entries from +the theoretical level. +3.1. +System Model +Definition 1. (Security Policy): +The high-level security policy is defined as +a finite set of access control relationships: SPM = { R0, R1,...,Rn | ∀ Ri = ( +si, oj, a ) }, where si ∈ S represents the subject of the relationship, oj ∈ O +represents the object of the relationship, a represents the access authorization, +i.e., the subject can access the object. +Definition 2. (Host): +The host existed in the data plane is defined as a tuple: +hi=( ipi, swi, portm +i +), where ipi represents the host’s IP address in the data +plane, swi represents the OpenFlow switch connected with the host, portm +swi +represents the port connected with the host in swi. +Definition 3. (OpenFlow Switch): +The OpenFlow switch existed the data +plane is defined as a finite set of flow entries: swi = { f0, f1, ..., fn }. +Definition 4. (Flow Entry): +The flow entry existed in the OpenFlow switch +is defined as a tuple: fi = ( ipsrc, ipdst, portin +swi =⇒ portout +swi ), where ipsrc +represents the traffic’s source IP address, ipdst represents the traffic’s destination +IP address, portin +swi =⇒ portout +swi represents the traffic input from portin +swi will be +outputted from portout +swi in the switch swi. +Definition 5. (Edge): +The edge existed in the data plane is defined as: ei += portout +start �−→ portin +end, where portout +start represents the port connected with the +edge in the start switch swstart, portin +end represents the port connected with the +edge in the end switch swend, so that the direction of the edge is from swstart +to swend. +Definition 6. (Topology): +The topology of the data plane is defined as an +graph: G = ( SW, E ), where SW represents a finite set of OpenFlow switches, +E represents a finite set of edges. +6 + +Definition 7. (Connected Path): +The connected path between the host hi +and the host hj in the topology G is defined as: Pi = hi �−→ hj = { hi, e0, e1,..., +en, hj | ∀ei ∈ E }, where the direction of the path is from hi to hj. +3.2. Transforming SPM into the Flow Entries +Based on the established system model, we propose a formal method to +transform SPM into the system model of flow entries from the theoretical level. +The method can be described as Figure 1 and summarized as follows: First of all, +for ∀Ri ∈ SPM, the subject si ∈ Ri is transformed into a host h(si) in the data +plane, the object oj ∈ Ri is transformed into a host h(oi) in the data plane; +Next, the access authorization a ∈ Ri is transformed into a connected path +Pi between h(si) and h(oj); After that, the connected path Pi is transformed +into a set of flow entries used by the switches which are passed by Pi; Finally, +SPM is transformed into the set of flow entries ∆ when all relationships of +SPM have been transformed. +As shown in Figure 1, the relationship R1 ∈ +SPM is transformed into the connected path P1 between h1 and h2, thus P1 +is transformed into the flow entries deployed in sw1, so as to implement the +security policy transformation from the system model level. The soundness of +the formal method has been proven in our previous paper[7]. That is, if the +security properties defined by SPM is denoted as ϕ, the system model of data +plane is denoted as D, the flow entries generated by the method is denoted as ∆, +then the method can ensure the data plane D loaded with ∆ can synchronously +and continuously hold the security properties ϕ at runtime, i.e., D(∆) |= ϕ. +Specifically, the formal method to transform SPM into the system model of +flow entries is defined as follows. +• Transforming the Subject Given an access control relationship Ri ∈ +SPM and the subject si ∈ Ri, then si is transformed into a corresponding +host existed in the data plane. The rule of transforming the subject of SPM is +formally defined as follows: +si ∈ Ri +h(si) = (ipsrc, swsrc, portin +src, ) +(1) +where h(si) represents the host mapped with si in the data plane, ipsrc repre- +sents the IP address of h(si) in the data plane, swsrc represents the OpenFlow +switch connected with h(si) in the data plane, portin +src represents the port con- +nected with h(si) in swsrc. +• +Transforming the Object Given an access control relationship Ri ∈ +SPM and the object oj ∈ Ri, then oj is transformed into a corresponding host +existed in the data plane. The rule of transforming the object of SPM is formally +defined as follows: +oj ∈ Ri +h(oj) = (ipdst, swdst, portout +dst, ) +(2) +where h(oj) represents the host mapped with oj in the data plane, ipdst repre- +sents the IP address of h(oj) in the data plane, swdst represents the OpenFlow +7 + +Figure 1: +The method of transforming SPM into the system model of flow entries. +8 + +- +Si +R1 +S +0 +Application PN +h3 +P1 +sw1 +h1 +h2 +sw2switch connected with h(oj), portout +dst represents the port connected with h(oj) +in swdst. +• Transforming the Authorization Given an access control relationship +Ri ∈ SPM and the access authorization a ∈ Ri, if there existing a connected +path Pi between h(si) and h(oj) in the topology, then a is transformed into the +connected Pi. The rule of transforming the authorization is formally defined as +follows: +a ∈ Ri +h(si) �−→ h(oj) ⊂ G +(3) +where h(si) �−→ h(oj) represents a directional connected path from h(si) to +h(oj) in the topology. +• Transforming the Path The connected path h(si) �−→ h(oj) is trans- +formed into a set of flow entries deployed in the OpenFlow switches which are +passed by the path. The rule of transforming the connected path is formally +defined as follows: +h(si) �−→ h(oj) +dst +� +k=src +fi(k) +(4) +where fi(k) =( ipsrc, ipdst, portin +k =⇒ portout +k +) represents a flow entry deployed +in the switch swk which is passed by h(si) �−→ h(oj). Leveraging the definitions +of the system model, we can proof that the rule of transforming the connected +path is sound. +proof : +h(si) �−→ h(oj) += { h(si), e0, e1, ..., en, h(oj) } += {(ipsrc, swsrc, portin +src), (portout +src �−→ portin +0 ) +, ..., (portout +n +�−→ portin +dst), (ipdst, swdst, portout +dst)} += {(portin +src =⇒ portout +src), (portin +0 =⇒ portout +0 +), +, ..., (portin +n =⇒ portout +n ), (portin +dst =⇒ portout +dst) +(ipsrc, ipdst), (swsrc, swdst)} += { +dst +� +k=src +(portin +k =⇒ portout +k )} � (ipsrc, ipdst) += +dst +� +k=src +{(portin +k =⇒ portout +k ) � (ipsrc, ipdst)} += +dst +� +k=src +fi(k). +Therefore, ∀Ri ∈ SPM, the subject si ∈ Ri, the object oj ∈ Ri and the +autherization a ∈ Ri, if ∃Pi = h(si) �−→ h(oj) in the topology, then Ri can +be transformed into a corresponding set of flow entries +dst +� +k=src +fi(k) by using the +system model step by step, it can be formally defined as follows: +Ri +dst +� +k=src +fi(k) +(5) +9 + +• +Transforming SPM SPM is transformed into a corresponding set of +flow entries ∆ by using the equation (5). If || SPM || = m, then the rule of +transforming SPM is formally defined as follows: +∆ = +m +� +i=1 +{ +dst +� +k=src +fi(k)} +(6) +4. The Security Policy Transformation Framework +The problem of how to transform SPM into the corresponding flow entries +used by the OpenFlow switches has been solved from the theoretical level in +Section 3. However, we cannot solve the problem of how to find a connected +path in the data plane for each relationship Ri ∈ SPM. In addition, by means +of the security policy transformation method, SPM can be transformed into the +corresponding flow entries, but the flow entry transformed from SPM is of the +system model, i.e. it is only the formal description of the real flow entry and +cannot be used by the real OpenFlow switch directly, so that we need to further +solve the problem of how to generate the practical flow entries by using the +system model of flow entries. Based on this insight, a runtime security policy +transformation framework for SDN networks is proposed from the practical level +in this section. By means of the framework, we can solve the problem of how +to find the connected path for each relationship defined by SPM, as well as +the problem of how to generate the practical flow entries based on the system +model at runtime. As shown in Figure 2, this framework consists of 5 functional +modules, i.e., the security policy module, the topology discovery module, the +runtime monitoring module, the path generation module and the flow entry +generation module. +4.1. Overview of the Framework +• Security policy module is deployed in the application plane and responsible +for maintaining the security policy (SPM). Each access control relationship Ri ∈ +SPM is designed as a 3-tuple: ( si, oj, fixed ) based on the system model, where +si is the subject; oj is the object; fixed={ 0, 1 } is a tag bit, fixed=1 represents +the relationship has been updated by user, fixed=0 represents it is unchanged. +SPM is stored as a text document and can be updated by the user at runtime. +• Topology discovery module is deployed in the control plane and responsible +for creating a dynamic real-time topology of the entire data plane by capturing +the link events transmitted from the OpenFlow switches. Based on the system +model, each edge in the topology is designed as a tuple: ei = ( swsrc, portsrc, +swdst, portdst, using, c ), where using={ True, False } is a tag bit and can +be changed by the real-time link events at runtime, using=True represents the +edge can be used now, using=False represents the edge is interrupted now. For +the convenient of researching, the cost of each edge is set to 1, i.e., c =1. The +generated topology is stored as a text document and can also be updated by +the real-time link events at runtime. +10 + +Figure 2: +The security policy transformation framework for SDN networks. +• Runtime monitoring module is deployed in the control plane and respon- +sible for monitoring all the traffics in the data plane by capturing the real-time +packet-in events transmitted from the OpenFlow switches. When a new packet- +in event arrives in the controller, the module first invokes the path generation +module to transform the latest security policy (SPM) into a set of connected +paths in the data plane, then invokes the flow entry generation module to trans- +form all the connected paths into their corresponding flow entries deployed in +the OpenFlow switches which are passed by these paths. As SPM and the topol- +ogy of data plane will be evolved with the runtime environment, this module is +designed to be triggered by the real-time packet-in events continuously, so that, +when SPM is changed (i.e., occurring fixed=1) or the topology is changed (i.e., +occurring using=False) at runtime, the module will first delete all the current +flow entries deployed in the OpenFlow switches, then update all the flow tables +by using the latest generated flow entries, so as to ensure the security properties +defined by SPM can be synchronously and continuously hold in the data plane +at runtime. +• Path generation module is deployed in the control plane and invoked by the +runtime monitoring module at runtime. The module is responsible for trans- +forming each Ri ∈ SPM input from the runtime monitoring module into a +corresponding path Pi in the data plane by using the latest topology file and +the path searching algorithm. Specifically, it first transforms the subject si ∈ Ri +and the object oj ∈ Ri into the hosts h(si) and h(oj) in the data plane respec- +11 + +Security Policy +Update +SPM +Module +Application Plane +SPM +Path +Topology +-Topology +Generation +Module +Control Plane +Update +Invoke +Path +Topology +Flow Entry +Runtime +Discovery +Generation +Path +Monitoring +Module +Module +Module +Link Events +Flow Entries +Packet-In Events +OpenFlow Switches +Data Planetively, then finds a shortest connected path Pi between h(si) and h(oj) by using +the path searching algorithm, finally all the connected paths transformed from +SPM are returned to the runtime monitoring module. +• Flow entry generation module is also deployed in the control plane and in- +voked by the runtime monitoring module at runtime. The module is responsible +for transforming the connected path into a set of flow entries deployed in the +OpenFlow switches which are passed by the path, then utilizing the instructions +provided by the controller to generate the practical flow entries and distributing +these flow entries to the corresponding OpenFlow switches at runtime. +4.2. Runtime Monitoring Algorithm +The runtime monitoring algorithm deployed in the runtime monitoring mod- +ule plays the role of coordinator in the framework, and can be described as Al- +gorithm 1 in pseudo code. First of all, the algorithm creates two dynamic lists +S and T by reading the latest SPM and Topo file respectively. If there existing +an access control relationship has been changed by user (Ri.fixed=1) or a edge +has been shut down in the topology (ei.using=False) at runtime, it will clear +all the current flow entries deployed in the OpenFlow switches for ready of the +updating. In the following, for each access control relationship Ri ∈ S, it maps +the subject si ∈ Ri and the object oj ∈ Ri with the switches swsrc and swdst +in the data plane, then transforms Ri into a corresponding connected path Pi +by invoking the path searching algorithm djk-route(swsrc, swdst, N). Based +on the transformation rules, Pi can be further transformed into a set of flow +entries. When all the relationships in the List S having been transformed, SPM +has been transformed into a corresponding set of flow entries ∆, the algorithm +invokes the flow entry generation module to update the data plane by using +∆. Since the algorithm is designed to be triggered by the packet-in events at +runtime, so that it ensures the framework can perceive any changes in time +when the security policy or the topology of data plane has been evolved with +the environment, and then update the data plane synchronously at runtime. +4.3. Path Searching Algorithm +Another important algorithm in the framework is the path searching algo- +rithm deployed in the path generation module. The algorithm is improved from +the classic Dijkstra algorithm and can be described as the Algorithm 2 in pseudo +code. First of all, the algorithm creates a dynamic matrix djk[N][N] by using +the sum of Openflow switches N and the topology file. In the following, it cal- +culates a shortest connected path between swsrc and swdst in the data plane +by using djk[N][N] and the created stacks. After multi-round calculating, the +shortest path Pi between swsrc and swdst is found and returned to the runtime +monitoring module. As the cost of each edge has been set to 1 and not consider- +ing of the quality of services (QoS) of edges, so that the shortest path Pi found +by the algorithm is generated by calculating the minimum number of hops in +the topology. Moreover, since the matrix djk[N][N] is dynamically created by +the topology file, so that the searched shortest path will be evolved with the +changing of the topology at runtime. +12 + +5. Implementation and Evaluations +In order to validate the feasibility and effectiveness of the framework pro- +posed in Section 4, we set up an experimental system and implement the frame- +work with POX controller[27] and Mininet emulator[28]. First of all, we im- +plement a virtual SDN network by using the Mininet emulator. As shown in +Figure 3, the topology of the network consists of 6 hosts (h1 ∼ h6) and 11 Open- +Flow switches (sw1 ∼ sw11). We further implement the security policy module, +topology discovery module, runtime monitoring module, path generation mod- +ule and flow entry generation module with Python 3.6.1 and integrate these +modules with the core of POX controller. The experimental system consists of +a Lenovo workstation with Windows OS, Intel-i7 32Cores 2.60GHz CPU, 32GB +RAM and a Raspberry platform with Linux OS, ARM-v7 CPU and 945MB +RAM. The POX controller and the functional modules are deployed in Lenovo +workstation, Mininet emulator is deployed in Raspberry platform, and Lenovo +workstation is connected with Raspberry platform using coaxial cable directly. +Table 1: The high-level security policy (SPM) +R1 +( 1, 5, 1 ) +R2 +( 5, 1, 1 ) +R3 +( 2, 4, 1 ) +R4 +( 4, 2, 1 ) +5.1. Effectiveness Evaluation +The security policy used for validating the effectiveness of the framework is +shown in Table I. Since any effective interaction is bidirectional in SDN networks, +i.e., the subject’s host and the object’s host must be ensured they can access +each other in the data plane, so that we design the security policy as 4 access +control relationships (R1 ∼ R4) to ensure h1 (1) and h5 (5) can access each +other, h2 (2) and h4 (4) can access each other, and all the relationships of +SPM are set as fixed=1, i.e., having been updated by user. In the following, +the effectiveness evaluations towards the framework will be carried out under 4 +different scenarios at runtime, they are the effectiveness after loading the flow +entries, the effectiveness after cutting down the path and the effectiveness after +changing SPM. +5.1.1. Effectiveness after Loading the Flow Entries +The purpose of this experiment is to validate whether the data plane after +loading the generated flow entries can hold the security properties defined by +SPM. First of all, the subjects and objects of SPM shown in Table I, i,e., the +1, 2, 4 and 5, are transformed into their corresponding hosts in the data plane +by using security policy transformation. Specifically, the 1 is transformed into +13 + +Figure 3: +The topology of the virtual SDN network. +h1=( 10.0.0.1, sw1, 1 ), the 2 is transformed into h2=( 10.0.0.2, sw2, 1 ), the 4 +is transformed into h4=( 10.0.0.4, sw9, 1 ) and the 5 is transformed into h5=( +10.0.0.5, sw10, 1 ) respectively. In the following, the path searching algorithm, +i.e., Algorithm 2, searches the shortest path between the subject’s host and the +object’s host for each Ri ∈ SPM based on the latest topology generated from +the topology file. +After that, the relationships shown in Table I have been transformed into 4 +corresponding shortest connected paths (P1 ∼ P4) in the data plane. As shown +in Figure 4, the R1 is transformed into P1={ h1, e1, e2, e3, h5 }, the R2 is +transformed into P2={ h5, e3, e2, e1, h1 }, the R3 is transformed into P3={ h2, +e4, e5, h4 } and the R4 is transformed into P4={ h4, e5, e4, h2 }, where the +P1 and P2 are depicted with the blue lines, the P3 and P4 are depicted with +the orange lines. In the following, the flow entry generation module transforms +each path into a set of flow entries deployed in the switches passed by the path. +Specifically, P1 and P2 are transformed into the flow entries deployed in the +switches { sw1, sw5, sw8, sw10 }, while P3 and P4 are transformed into the flow +entries deployed in the switches { sw2, sw6, sw9 }. +After that, we execute the pingall instruction in the Mininet CLI and observe +the reachability of the entire data plane. As shown in Figure 5, h1 and h5 can +access each other, h2 and h4 can access each other either, so that the data plane +after loading the generated flow entries has been proven that it can hold all the +security properties defined by SPM. +14 + +Figure 4: +The shortest paths searched by Algorithm 2. +Figure 5: +The result of executing the pingall instruction. +15 + +1 +e +es +e3pi@raspberrypi:/mininet/custom +口 +X +s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 +*** Adding links: +(hl, sl) (h2, s2) (h3, s7) (h4, s9) (h5, s10) (h6, sl1) (sl, s2) (sl, s3) (sl, s +5) (s2, s4) (s2, s6) (s3, s4) (s3, s7) (s4, s5) (s4, s7) (s4, s8) (s5, s6) (s5, +s8) (s5. s9) (s6, s9) (s7. s8) (s7. s10) (s8. s9) (s8. s10) (s8. s11) (s9, s11) +(s10, sl1) +*** Configuring hosts +hl h2 h3 h4 h5 h6 +*** starting controller +c0 +*** starting ll switches +s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 +*** Starting CLI: +mininet> pingall +*** Ping: testing ping reachability +hl +-> X X X h5 X +h2 +-> x x h4 x x +13 +XXXXX<- +h4 -> x h2 X X +n5 -> hl x x x X +X X X X X <- 9u +*** Results: 86% dropped (4/30 received) +mininet> +mininet>5.1.2. Effectiveness after Cutting down the Path +The purpose of this experiment is to validate whether the security properties +defined by SPM can be continuously hold in the data plane after the connected +path between the subject’s host and the object’s host is shut down at runtime. +If the framework can synchronously perceive this change from the topology +and automatically find another new connected path to keep the data plane +holding the security properties at runtime, then the framework will be proven +to be effective under this scenario. First of all, we let h1 and h5 can access +each other by loading the corresponding flow entries into the data plane, and +the shortest path between h1 and h5 is initialed with { h1, e1, e2, e3, h5 }. +In the following, we make a continuous TCP traffic sent from h1 to h5 by +using the iperf instruction in the Mininet CLI, we set the duration time of the +experiment equals 60 seconds and record the throughput of the traffics in h5. +When the time reaches 23 seconds, we shut down the edge between sw8 and +sw10 existed in the path P1 by using the instruction in the Mininet CLI and let +the experiment going on. When the experiment is finished, we read the data +recorded in h5 and plot them in Figure 6. As shown in Figure 6, the throughput +of the traffic sent from h1 to h5 is sharply declined after the edge is shut down at +23 seconds, and completely becomes zero from 25 seconds to 37 seconds. After +38 seconds, the traffic quickly returns to normal until the end of the experiment. +The experimental result has illustrated that the framework can synchronously +perceive the change caused by cutting down one edge between sw8 and sw10, +and automatically find another new shortest connected path between h1 and +h5, i.e., { h1, e6, e7, e8, h5 }, so as to make the traffic returning to normal +quickly and keep the data plane holding the security properties defined by SPM +at runtime. The new shortest connected path searched by Algorithm 2 after +cutting down the using path is shown in Figure 7. +5.1.3. Effectiveness after Changing SPM +The purpose of this experiment is to validate whether the security properties +defined by SPM can be continuously hold in the data plane after SPM is changed +by the user at runtime. If the framework can synchronously perceive this change +from SPM and automatically update the flow entries deployed in the switches +to keep the data plane holding the evolved security properties at runtime, then +the framework will be proven to be effective under this scenario. First of all, +we let h1 and h5 can access each other by loading a corresponding flow entries +into the data plane, then make a continuous TCP traffic sent from h1 to h5 by +using the iperf instruction in the Mininet CLI, we set the duration time of the +experiment equals 60 seconds and record all the throughput data of the traffic +in h5. +In the following, we first validate the effectiveness of the framework under +the scenario of changing SPM from h1 and h5 can access each other to h2 and h5 +can access each other at runtime, and the experimental result under this scenario +is plotted in Figure 8. As shown in Figure 8, the throughput of the traffic sent +from h1 to h5, which is depicted with the red line, is quickly declined when we +load the new SPM into the controller at 29 seconds, and completely becomes +16 + +Figure 6: +The blue line represents the throughput of the traffic sent from h1 to h5. The +dotted line represents we shut down the connected path at 23 seconds. +Figure 7: +The new connected path searched by Algorithm 2 at runtime. +17 + +Throughput (Gbits/sec) +the traffic from h1 to h5 +-shut down the path at runtime +20 +60 +Time (sec)e6 +eA +1 +e7 +es +es +eszero after 31 seconds. From 31 seconds until to the end of the experiment, h5 +can only receive the traffic sent from h2 which is depicted with the blue line. +The experimental result illustrates the framework can synchronously perceive +this change and keep the data plane holding the evolved security properties after +changing SPM from h1 and h5 can access each other to h2 and h5 can access +each other at runtime. +Figure 8: +The red line represents the throughput of the traffic sent from h1 to h5, the blue +line represents the throughput of the traffic sent from h2 to h5, the dotted line represents we +change SPM at 29 seconds. +After that, we further validate the effectiveness of the framework under the +scenario of adding a new relationship into SPM, i.e., adding h2 and h5 can access +each other, at runtime. The experimental result under this scenario is plotted +in Figure 9. As shown in Figure 9, the throughput of the traffic sent from h1 to +h5, which is depicted with the red line, still keeps normal before we load the new +SPM into the controller at 43 seconds. From 44 seconds until to the end of the +experiment, h5 can receive the continuous traffic sent from h2 which is depicted +with the blue line, and can also receive the traffic sent from h1 at the same time. +Due to the crowding of the traffic sent from h2, the throughput of the traffic +from h1 is declined from 3.71GB/s to 1.99 GB/s. The throughput of the traffic +sent from h2 is still kept between 1.6GB/s and 2.2GB/s after 44 seconds. The +experimental result illustrates the framework can synchronously perceive this +change and keep the data plane holding the evolved security properties after +adding a new relationship into SPM at runtime. +18 + +Throughput (Gbits/sec) +the traffic from h1 to h5 +the traffic from h2 to h5 +-load the new SPM at runtime +Time (sec)Figure 9: +The red line represents the throughput of the traffic sent from h1 to h5, the blue +line represents the throughput of the traffic sent from h2 to h5, the dotted line represents we +change SPM at 43 seconds. +5.2. Performance Evaluation +As the critical algorithm used for implementing the security policy trans- +formation, the performance of the path searching algorithm, i.e., Algorithm 2, +needs to be further evaluated. First of all, the sum of access control relation- +ships of the security policy (SPM) is denoted as M, and the sum of OpenFlow +switches in the topology is denoted as N in this performance evaluation. Then +by leveraging the Python programming, the execution time of Algorithm 2 have +been recorded in milliseconds (ms) for calculating the shortest paths under set- +ting the different value of M and N. +The experimental result is plotted in +Figure 10. As shown in Figure 10, with gradually amplifying the value of M +from 2 to 10, and the value of N from 11 to 400 respectively, the execution time +of Algorithm 2 shows an obvious exponential upward trend. Moreover, accord- +ing to the description of Algorithm 2, the time complexity for calculating only +one shortest path will reach O(N 2), because the Algorithm 2 needs to create +a dynamic matrix djk[N][N] and further calculates the while loop, so that the +time complexity for transforming all the access control relationships defined by +SPM into their corresponding shortest paths will reach O(M × N 2). +6. Conclusion +In this paper, we propose a practical runtime security policy transforma- +tion framework for SDN networks. First of all, we specify the security policies +used by SDN networks, such as access control policies or firewall policies, as a +19 + +Throughput (Gbits/sec) +the traffic from h1 to h5 +一the traffic from h2 to h5 +-load the new SPM at runtime +Time (sec)Figure 10: +The execution time of Algorithm 2 recorded in milliseconds (ms) for calculating +the shortest paths under setting the different value of M and N, where M represents the sum +of access control relationships of SPM, N represents the sum of OpenFlow switches in the +topology. +system model of security policy (SPM). SPM is of a high-level system model +without containing any underlying information of data plane. From the theoret- +ical level, we establish the system model for SDN network and propose a formal +method to transform SPM into the corresponding flow entries automatically. +The flow entry transformed from SPM is of a low-level system model containing +the underlying information of data plane. From the practical level, we propose +a runtime security policy transformation framework which consists of the se- +curity policy module, topology discovery module, runtime monitoring module, +path generation module, as well as flow entry generation module. Leveraging +these functional modules, the framework can solve the problem of how to find +a connected path for each relationship defined by SPM in the data plane, how +to transform the path into the system model of flow entries, as well as how to +generate the practical flow entries by using the system model of flow entries. +In order to validate the feasibility and effectiveness of the framework, we set +up an experimental system and implement the framework by using POX con- +troller and Mininet emulator. The experimental result illustrate the framework +is completely effective at runtime. +However, there still exists some problems needed to be further researched +in the future. The current path searching algorithm, i.e., Algorithm 2, used +by the framework is improved from the classic Dijkstra algorithm and finds +the shortest path by calculating the minimum number of hops in the topology. +However, in the real SDN networks, the problem of searching a connected path +between the two hosts need to consider the matters of quality of service (QoS), +20 + +350 +when M =2 +when M =4 +M +300- +when M =6 +when M =8 +when M =10 +250 +(sw) +e 200 +Performance +W +150 +P +100 - +50 - ++0 +0 +50 +100 +150 +200 +250 +300 +350 +400 +Sum of OpenFlow Switches (N)load balance, as well as some specific requirements about the traffic engineering +at runtime, so that the framework needs to be further improved by employing +some novel path searching algorithms based on multi-object optimization[29] or +reinforcement learning[30] methods. +Acknowledgment +This paper has been sponsored and supported by National Key Research +and Development Program of China (Grant No.2018YFB0803400), Doctoral +Foundation of Qingdao Binhai University (Grant No. BS2022A10), partially +supported by Key Program of National Natural Science Foundation of China +(Grant No.61932013). +References +References +[1] D. 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Serrat, Finite state transducers for policy evaluation and +conflict resolution, in: IEEE International Workshop on Policies for Dis- +tributed Systems and Networks, 2004, pp. 250–259. +[27] POX (2017). [link]. +URL https://github.com/noxrepo/pox +[28] Mininet (2012). [link]. +URL http://www.mininet.org +[29] R. T. Marler, J. S. Arora, Survey of multi-objective optimization methods +for engineering, Structural and multidisciplinary optimization 26 (6) (2004) +369–395. +[30] L. P. Kaelbling, M. L. Littman, A. W. Moore, Reinforcement learning: A +survey, Journal of artificial intelligence research 4 (1996) 237–285. +23 + diff --git a/DdE2T4oBgHgl3EQfSQc5/content/tmp_files/load_file.txt b/DdE2T4oBgHgl3EQfSQc5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e7e86b56fe49043c98c2db4ace26d49b417c955a --- /dev/null +++ b/DdE2T4oBgHgl3EQfSQc5/content/tmp_files/load_file.txt @@ -0,0 +1,553 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf,len=552 +page_content='A Practical Runtime Security Policy Transformation Framework for Software Defined Networks Yunfei Menga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Changbo Keb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Zhiqiu Huangc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Guohua Shenc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' ChunQiang Liua,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Xiaojie Fenga aCollege of Information Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Qingdao Binhai University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Qingdao 266555,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' China bSchool of Computer Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Nanjing University of Posts and Telecommunications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Nanjing 210023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' China cCollege of Computer Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Nanjing University of Aeronautics and Astronautics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Nanjing 211106,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' China Abstract Software-defined networking (SDN) has been widely utilized to enforce the se- curity of traditional networks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' thereby promoting the process of transforming traditional networks into SDN networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' However, SDN-based security enforce- ment mechanisms rely heavily on the security policies containing the underlying information of data plane, such as MAC address, IP address or switch ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' These security policies need to be specifically developed by the network opera- tors, and loaded into the control plane by manual inputting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' With increasing the scale of underlying network, the current security policy management mechanism will confront more and more challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The security policy transformation for SDN networks is to research how to transform the high-level security policy without containing the underlying information of data plane into the practical flow entries used by the OpenFlow switches automatically, thereby implement- ing the automation of security policy management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Based on this insight, a practical runtime security policy transformation framework is proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' First of all, we specify the security policies used by SDN networks as a system model of security policy (SPM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' From the theoretical level, we establish the system model for SDN network and propose a formal method to transform SPM into the system model of flow entries automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' From the practical level, we propose a runtime security policy transformation framework to solve the problem of how to find a connected path for each relationship of SPM in the data plane, as well as how to generate the practical flow entries according to the system model of flow entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In order to validate the feasibility and ef- fectiveness of the framework, we set up an experimental system and implement the framework with POX controller and Mininet emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The experimental results illustrate the framework can synchronously perceive the changes caused by cutting down one edge or changing SPM, and keep the data plane holding the security properties defined by SPM continuously at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Keywords: SDN, security policy, model transformation, data plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Preprint submitted to Elsevier January 11, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='03790v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='CR] 10 Jan 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Introduction Software-defined networking (SDN) is a novel networking technique or ar- chitecture that changes the limitation of traditional network infrastructures by breaking the vertical integration, decoupling the control logics from the underly- ing forwarding devices, promoting the centralization of control and introducing the abilities to program the network directly[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In SDN networks, the control logics of network, such as routing, traffic engineering or security policy devel- oped in the application plane, are loaded into the control plane via the north- bound interfaces (NBI) and transformed into a set of forward entries used by the OpenFlow switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' After that, the control plane distributes the generated flow entries to the associated switches in the data plane via the southbound interfaces (SBI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Because of its programmable, centralized intelligent control as well as global traffic view, SDN has been widely utilized to enhance the security of tradition networks, thereby promoting the process of transforming traditional networks into SDN networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' For instances, Garay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' [2] proposed a SDN-based network access control mechanism, flownac, which is a centralized EAP (extensible authentication protocol) for IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='1x wireless local area network (WLAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Yakasai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' [3] proposed a network access control mech- anism, flowidentity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' This mechanism integrates EAP security authentication mechanism into the SDN controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' [4] proposed a dynamic firewall mechanism, flowguard, based on SDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Koerner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' [5] proposed a device se- curity authentication mechanism based on MAC address and SDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' However, SDN-based security enforcement mechanisms rely heavily on the security policies containing the underlying information of the data plane, such as MAC address, IP address or switch ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' These security policies need to be specifically developed by the network operators, and loaded into the control plane by means of the manual inputting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' With increasing the scale of underlying network, the current security policy management mechanism will confront more and more challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' First of all, it is nearly impossible for any operator to completely understand all the information of underlying network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In addition, with the emergence of multi-controller SDN[6], network operators need to man- age a variety of heterogeneous controllers at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In this case, the same security policy often needs to be developed and deployed for the different types of controller, which inevitably increases the complexity and difficulty for network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Therefore, a novel security policy management mecha- nism which can be completely transparent to the underlying information of data plane is urgently needed for SDN networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' That is, it can permit the operators only to define the high-level security policy without containing any underlying information, then by means of the security policy transformation, the high-level security policy can be automatically transformed into its corresponding flow entries used by the OpenFlow switches in the data plane, thereby implementing the automation of security policy management for SDN networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 2 Based on these insights, we have proposed a security policy model transfor- mation and verification approach for SDN networks and published the approach in our previous paper[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In that paper, we proposed a security policy transfor- mation method to transform the high-level security policy model (SPM) without containing the underlying information into its corresponding low-level security policy model (LSPM) containing the underlying information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' To verify the soundness of proposed security policy model transformation method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' we further proposed a security policy verification method and proved that the problem of whether the data plane can satisfy the security properties defined by SPM is equivalent to the problem of searching the connected paths related with SPM in the data plane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' as long as each access control relationship Ri ∈ SPM can be transformed into a corresponding connected path Pi in the data plane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' next transforms Pi into a set of flow entries used by the OpenFlow switches,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' then the data plane must can hold the security properties defined by SPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' However, that paper only proposed the method from the theoretical level, and did not specifically implement this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Moreover, it did not solve the problem of how to find a connected path for each relationship of SPM in the data plane, and how to transform LSPM into the practical flow entries used by the switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' And based on the theoretical foundation of that paper, we propose a run- time security policy transformation framework for SDN networks in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' First of all, this paper further improves the system model of SDN networks and solves the problem of how to transform SPM into the flow entries used by the OpenFlow switches from the theoretical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Moreover, this paper proposes a runtime security policy transformation framework from the practical level, thereby solving the problem of how to find a connected path Pi for each rela- tionship Ri ∈ SPM in the data plane, as well as how to transform the system model of flow entries into the practical flow entries used by the switches at run- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In addition, this paper further implements the proposed framework with an experimental system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The experimental result illustrate the framework is completely effective at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Hence, the contributions of this paper can be summarized as follows: We specify the security policies used by SDN networks, such as access control policies or firewall policies, as a system model of security policy (SPM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' SPM is of a high-level system model without containing any underlying infor- mation of data plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' From the theoretical level, we establish the system model for SDN network, and propose a formal method to transform SPM into the system model of flow entries automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The system model of flow entry is of a low-level system model containing the underlying information of data plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' From the practical level, we propose a runtime security policy transforma- tion framework which consists of the security policy module, topology discovery module, runtime monitoring module, path generation module and flow entry generation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Leveraging these functional modules, the framework can solve the problem of how to find a connected path for each relationship of SPM in the data plane, how to transform the path into the system model of flow en- tries, as well we how to generate the practical flow entries by using the system 3 model of flow entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In order to validate the feasibility and effectiveness of the framework, we set up an experimental system and implement the framework by using POX con- troller and Mininet emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The experimental result illustrate the framework is completely effective at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The remainder of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Section 2 discusses some related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Section 3 proposes the system model and elaborates on how to transform SPM into the flow entries from the theoretical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Sec- tion 4 proposes the runtime security policy transformation framework from the practical level and introduces its functional modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Section 5 implements the framework with an experimental system and elaborates on how to evaluate the effectiveness and performance of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Finally, Section 6 concludes this paper and presents some future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Related Work In this section, we discuss some research works related with the policy model transformation and the security policy verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Policy Model Transformation According to the definitions of model-driven architecture (MDA), the model transformation refers to the process of transforming the platform independent model (PIM) to its corresponding platform specific model (PSM)[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' As far as the literatures we have read, the researches towards the policy model transforma- tion can be roughly divided into three categories, they are the template-based transformation, RBAC-oriented transformation as well as the transformation based on the system model and mapping rules[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Due to the limitation of tem- plate, the template-based model transformation has very limited transformation capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Generally, RBAC-oriented model transformation[10] is only suitable for transforming RBAC (role-based access control) policies, and does not have enough capability to describe the complex system, so that these two methods are not suitable for SDN networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' At present, the model transformation based on the system model and map- ping rules has been widely used for transforming the policy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The main idea of this method can be summarized as follows: (1) System Model: it de- fines the objects of system and the relationship between the system objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' (2) Policy Model: it defines the policy object and the relationship between the policy objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' (3) Mapping Rules: it establishes the mapping rules between the upper-level policy objects and the lower-level system objects[11][12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The trans- formation based on the system model and mapping rules first establishes the policy model and the system model which can describe the underlying system, then establishes the mapping rules between the policy objects and the system objects, then transforms the upper-level policy model into its corresponding lower-level policy model by means of the established mapping rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In partic- ular, Davy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' [13] proposed a policy model transformation method based on 4 mapping rules, in which the policy model is defined as a tuple ( event, condi- tion, behavior, subject, object ) and used the ontology to establish the mapping rules between the different system layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Luck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' [14] proposed a method to transform RBAC model defined in service layer into the policy model used in the system layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In this method, the system model is divided into three layers: roles and object (RO), subject and resources (SR) and processes and hosts (PH), and the mapping rules between the three layers have been estab- lished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Based on the Luck’s research, Porto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' [15] further decomposes the PH layer into two sub layers, namely DAS (diagram abstract subsystem) layer and PH layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' DAS layer is mainly used to describe the network topology in the original PH layer, while PH layer is used to describe the specific network information in DAS layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In addition, the authors also proposed a policy ver- ification framework, which can be used to verify the consistency problems in the process of policy transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In addition, Lampson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' [16] proposed a network policy model transformation method for the distributed computing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Maullo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' [17] proposed a policy transformation system based on the first-order predicate logics, which transforms the high-level policy model into the low-level network configuration policy through the network topology and other information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Nanxi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' [18] proposed a SDN-oriented access control policy transformation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In this paper, In this paper, we also propose a security policy transformation framework based on the system model and map- ping rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' We first establish the system model of security policy (SPM) and data plane, then establish the transformation rules between the policy objects of SPM and the objects of the data plane, thereby transforming SPM into the system model of flow entries automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Security Policy Verification To assure the information systems running securely, security mechanisms of information system need to be validated whether it can satisfy the security properties defined by the security policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The traditional validation methods based on the testing and simulation can only confirm the system can work prop- erly under the different testing scenarios, but it is difficult to find some hidden scenarios that occur with little probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Formal verification methods have been applied to overcome the shortcomings existed in the traditional valida- tion methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' At present, the formal verification methods for validating the security policy mainly include theorem proving and model checking[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The- orem proving is unsuitable to validate the properties of complex systems due to its lower efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Model checking[20] can be used to validate whether the system model can satisfy the expected dynamic behaviors and specific static properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Model checking technique has been widely used for the security policy verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' For instances, Al-Shaer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' [21] proposed a static policy inconsistency detection method for the firewall policies of network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Bandara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' [22] proposed a security policy verification framework based on event calcu- lus (EC) and used the reasoning techniques for the policy conflict identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' May et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' [23] verified the privacy policies by means of an asynchronous model checker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Rubio-Loyola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' [24] proposed a goal-oriented policy refinement and 5 conflict detection method by means of the model checking technique and linear temporal Logic (LTL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Graham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' [25] proposed a policy conflict detection method with the model checking and an extended decision table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Baliosan and Serrat[26] proposed a specific finite automata based method for the policy con- flict detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Problem Formalization The security policy transformation for SDN networks is to research how to transform the high-level security policy without containing the underlying infor- mation into the set of practical flow entries used by the OpenFlow switches in the data plane automatically, thereby implementing the automation of security policy management in SDN network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In the following of this section, we first establish the system model for SDN network, then propose a formal method to transform the security policy (SPM) into the system model of flow entries from the theoretical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' System Model Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' (Security Policy): The high-level security policy is defined as a finite set of access control relationships: SPM = { R0, R1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=',Rn | ∀ Ri = ( si, oj, a ) }, where si ∈ S represents the subject of the relationship, oj ∈ O represents the object of the relationship, a represents the access authorization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', the subject can access the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' (Host): The host existed in the data plane is defined as a tuple: hi=( ipi, swi, portm i ), where ipi represents the host’s IP address in the data plane, swi represents the OpenFlow switch connected with the host, portm swi represents the port connected with the host in swi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' (OpenFlow Switch): The OpenFlow switch existed the data plane is defined as a finite set of flow entries: swi = { f0, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', fn }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' (Flow Entry): The flow entry existed in the OpenFlow switch is defined as a tuple: fi = ( ipsrc, ipdst, portin swi =⇒ portout swi ), where ipsrc represents the traffic’s source IP address, ipdst represents the traffic’s destination IP address, portin swi =⇒ portout swi represents the traffic input from portin swi will be outputted from portout swi in the switch swi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' (Edge): The edge existed in the data plane is defined as: ei = portout start �−→ portin end, where portout start represents the port connected with the edge in the start switch swstart, portin end represents the port connected with the edge in the end switch swend, so that the direction of the edge is from swstart to swend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' (Topology): The topology of the data plane is defined as an graph: G = ( SW, E ), where SW represents a finite set of OpenFlow switches, E represents a finite set of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 6 Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' (Connected Path): The connected path between the host hi and the host hj in the topology G is defined as: Pi = hi �−→ hj = { hi, e0, e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', en, hj | ∀ei ∈ E }, where the direction of the path is from hi to hj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Transforming SPM into the Flow Entries Based on the established system model, we propose a formal method to transform SPM into the system model of flow entries from the theoretical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The method can be described as Figure 1 and summarized as follows: First of all, for ∀Ri ∈ SPM, the subject si ∈ Ri is transformed into a host h(si) in the data plane, the object oj ∈ Ri is transformed into a host h(oi) in the data plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Next, the access authorization a ∈ Ri is transformed into a connected path Pi between h(si) and h(oj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' After that, the connected path Pi is transformed into a set of flow entries used by the switches which are passed by Pi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Finally, SPM is transformed into the set of flow entries ∆ when all relationships of SPM have been transformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' As shown in Figure 1, the relationship R1 ∈ SPM is transformed into the connected path P1 between h1 and h2, thus P1 is transformed into the flow entries deployed in sw1, so as to implement the security policy transformation from the system model level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The soundness of the formal method has been proven in our previous paper[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' That is, if the security properties defined by SPM is denoted as ϕ, the system model of data plane is denoted as D, the flow entries generated by the method is denoted as ∆, then the method can ensure the data plane D loaded with ∆ can synchronously and continuously hold the security properties ϕ at runtime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', D(∆) |= ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Specifically, the formal method to transform SPM into the system model of flow entries is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Transforming the Subject Given an access control relationship Ri ∈ SPM and the subject si ∈ Ri, then si is transformed into a corresponding host existed in the data plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The rule of transforming the subject of SPM is formally defined as follows: si ∈ Ri h(si) = (ipsrc, swsrc, portin src, ) (1) where h(si) represents the host mapped with si in the data plane, ipsrc repre- sents the IP address of h(si) in the data plane, swsrc represents the OpenFlow switch connected with h(si) in the data plane, portin src represents the port con- nected with h(si) in swsrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Transforming the Object Given an access control relationship Ri ∈ SPM and the object oj ∈ Ri, then oj is transformed into a corresponding host existed in the data plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The rule of transforming the object of SPM is formally defined as follows: oj ∈ Ri h(oj) = (ipdst, swdst, portout dst, ) (2) where h(oj) represents the host mapped with oj in the data plane, ipdst repre- sents the IP address of h(oj) in the data plane, swdst represents the OpenFlow 7 Figure 1: The method of transforming SPM into the system model of flow entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 8 Si R1 S 0 Application PN h3 P1 sw1 h1 h2 sw2switch connected with h(oj), portout dst represents the port connected with h(oj) in swdst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Transforming the Authorization Given an access control relationship Ri ∈ SPM and the access authorization a ∈ Ri, if there existing a connected path Pi between h(si) and h(oj) in the topology, then a is transformed into the connected Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The rule of transforming the authorization is formally defined as follows: a ∈ Ri h(si) �−→ h(oj) ⊂ G (3) where h(si) �−→ h(oj) represents a directional connected path from h(si) to h(oj) in the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Transforming the Path The connected path h(si) �−→ h(oj) is trans- formed into a set of flow entries deployed in the OpenFlow switches which are passed by the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The rule of transforming the connected path is formally defined as follows: h(si) �−→ h(oj) dst � k=src fi(k) (4) where fi(k) =( ipsrc, ipdst, portin k =⇒ portout k ) represents a flow entry deployed in the switch swk which is passed by h(si) �−→ h(oj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Leveraging the definitions of the system model, we can proof that the rule of transforming the connected path is sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' proof : h(si) �−→ h(oj) = { h(si), e0, e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', en, h(oj) } = {(ipsrc, swsrc, portin src), (portout src �−→ portin 0 ) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', (portout n �−→ portin dst), (ipdst, swdst, portout dst)} = {(portin src =⇒ portout src), (portin 0 =⇒ portout 0 ), , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', (portin n =⇒ portout n ), (portin dst =⇒ portout dst) (ipsrc, ipdst), (swsrc, swdst)} = { dst � k=src (portin k =⇒ portout k )} � (ipsrc, ipdst) = dst � k=src {(portin k =⇒ portout k ) � (ipsrc, ipdst)} = dst � k=src fi(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Therefore, ∀Ri ∈ SPM, the subject si ∈ Ri, the object oj ∈ Ri and the autherization a ∈ Ri, if ∃Pi = h(si) �−→ h(oj) in the topology, then Ri can be transformed into a corresponding set of flow entries dst � k=src fi(k) by using the system model step by step, it can be formally defined as follows: Ri dst � k=src fi(k) (5) 9 Transforming SPM SPM is transformed into a corresponding set of flow entries ∆ by using the equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' If || SPM || = m, then the rule of transforming SPM is formally defined as follows: ∆ = m � i=1 { dst � k=src fi(k)} (6) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The Security Policy Transformation Framework The problem of how to transform SPM into the corresponding flow entries used by the OpenFlow switches has been solved from the theoretical level in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' However, we cannot solve the problem of how to find a connected path in the data plane for each relationship Ri ∈ SPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In addition, by means of the security policy transformation method, SPM can be transformed into the corresponding flow entries, but the flow entry transformed from SPM is of the system model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' it is only the formal description of the real flow entry and cannot be used by the real OpenFlow switch directly, so that we need to further solve the problem of how to generate the practical flow entries by using the system model of flow entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Based on this insight, a runtime security policy transformation framework for SDN networks is proposed from the practical level in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' By means of the framework, we can solve the problem of how to find the connected path for each relationship defined by SPM, as well as the problem of how to generate the practical flow entries based on the system model at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' As shown in Figure 2, this framework consists of 5 functional modules, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', the security policy module, the topology discovery module, the runtime monitoring module, the path generation module and the flow entry generation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Overview of the Framework Security policy module is deployed in the application plane and responsible for maintaining the security policy (SPM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Each access control relationship Ri ∈ SPM is designed as a 3-tuple: ( si, oj, fixed ) based on the system model, where si is the subject;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' oj is the object;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' fixed={ 0, 1 } is a tag bit, fixed=1 represents the relationship has been updated by user, fixed=0 represents it is unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' SPM is stored as a text document and can be updated by the user at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Topology discovery module is deployed in the control plane and responsible for creating a dynamic real-time topology of the entire data plane by capturing the link events transmitted from the OpenFlow switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Based on the system model, each edge in the topology is designed as a tuple: ei = ( swsrc, portsrc, swdst, portdst, using, c ), where using={ True, False } is a tag bit and can be changed by the real-time link events at runtime, using=True represents the edge can be used now, using=False represents the edge is interrupted now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' For the convenient of researching, the cost of each edge is set to 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', c =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The generated topology is stored as a text document and can also be updated by the real-time link events at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 10 Figure 2: The security policy transformation framework for SDN networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Runtime monitoring module is deployed in the control plane and respon- sible for monitoring all the traffics in the data plane by capturing the real-time packet-in events transmitted from the OpenFlow switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' When a new packet- in event arrives in the controller, the module first invokes the path generation module to transform the latest security policy (SPM) into a set of connected paths in the data plane, then invokes the flow entry generation module to trans- form all the connected paths into their corresponding flow entries deployed in the OpenFlow switches which are passed by these paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' As SPM and the topol- ogy of data plane will be evolved with the runtime environment, this module is designed to be triggered by the real-time packet-in events continuously, so that, when SPM is changed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', occurring fixed=1) or the topology is changed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', occurring using=False) at runtime, the module will first delete all the current flow entries deployed in the OpenFlow switches, then update all the flow tables by using the latest generated flow entries, so as to ensure the security properties defined by SPM can be synchronously and continuously hold in the data plane at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Path generation module is deployed in the control plane and invoked by the runtime monitoring module at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The module is responsible for trans- forming each Ri ∈ SPM input from the runtime monitoring module into a corresponding path Pi in the data plane by using the latest topology file and the path searching algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' it first transforms the subject si ∈ Ri and the object oj ∈ Ri into the hosts h(si) and h(oj) in the data plane respec- 11 Security Policy Update SPM Module Application Plane SPM Path Topology Topology Generation Module Control Plane Update Invoke Path Topology Flow Entry Runtime Discovery Generation Path Monitoring Module Module Module Link Events Flow Entries Packet-In Events OpenFlow Switches Data Planetively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' then finds a shortest connected path Pi between h(si) and h(oj) by using the path searching algorithm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' finally all the connected paths transformed from SPM are returned to the runtime monitoring module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Flow entry generation module is also deployed in the control plane and in- voked by the runtime monitoring module at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The module is responsible for transforming the connected path into a set of flow entries deployed in the OpenFlow switches which are passed by the path, then utilizing the instructions provided by the controller to generate the practical flow entries and distributing these flow entries to the corresponding OpenFlow switches at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Runtime Monitoring Algorithm The runtime monitoring algorithm deployed in the runtime monitoring mod- ule plays the role of coordinator in the framework, and can be described as Al- gorithm 1 in pseudo code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' First of all, the algorithm creates two dynamic lists S and T by reading the latest SPM and Topo file respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' If there existing an access control relationship has been changed by user (Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='fixed=1) or a edge has been shut down in the topology (ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='using=False) at runtime, it will clear all the current flow entries deployed in the OpenFlow switches for ready of the updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In the following, for each access control relationship Ri ∈ S, it maps the subject si ∈ Ri and the object oj ∈ Ri with the switches swsrc and swdst in the data plane, then transforms Ri into a corresponding connected path Pi by invoking the path searching algorithm djk-route(swsrc, swdst, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Based on the transformation rules, Pi can be further transformed into a set of flow entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' When all the relationships in the List S having been transformed, SPM has been transformed into a corresponding set of flow entries ∆, the algorithm invokes the flow entry generation module to update the data plane by using ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Since the algorithm is designed to be triggered by the packet-in events at runtime, so that it ensures the framework can perceive any changes in time when the security policy or the topology of data plane has been evolved with the environment, and then update the data plane synchronously at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Path Searching Algorithm Another important algorithm in the framework is the path searching algo- rithm deployed in the path generation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The algorithm is improved from the classic Dijkstra algorithm and can be described as the Algorithm 2 in pseudo code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' First of all, the algorithm creates a dynamic matrix djk[N][N] by using the sum of Openflow switches N and the topology file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In the following, it cal- culates a shortest connected path between swsrc and swdst in the data plane by using djk[N][N] and the created stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' After multi-round calculating, the shortest path Pi between swsrc and swdst is found and returned to the runtime monitoring module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' As the cost of each edge has been set to 1 and not consider- ing of the quality of services (QoS) of edges, so that the shortest path Pi found by the algorithm is generated by calculating the minimum number of hops in the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Moreover, since the matrix djk[N][N] is dynamically created by the topology file, so that the searched shortest path will be evolved with the changing of the topology at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Implementation and Evaluations In order to validate the feasibility and effectiveness of the framework pro- posed in Section 4, we set up an experimental system and implement the frame- work with POX controller[27] and Mininet emulator[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' First of all, we im- plement a virtual SDN network by using the Mininet emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' As shown in Figure 3, the topology of the network consists of 6 hosts (h1 ∼ h6) and 11 Open- Flow switches (sw1 ∼ sw11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' We further implement the security policy module, topology discovery module, runtime monitoring module, path generation mod- ule and flow entry generation module with Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='1 and integrate these modules with the core of POX controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The experimental system consists of a Lenovo workstation with Windows OS, Intel-i7 32Cores 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='60GHz CPU, 32GB RAM and a Raspberry platform with Linux OS, ARM-v7 CPU and 945MB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The POX controller and the functional modules are deployed in Lenovo workstation, Mininet emulator is deployed in Raspberry platform, and Lenovo workstation is connected with Raspberry platform using coaxial cable directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Table 1: The high-level security policy (SPM) R1 ( 1, 5, 1 ) R2 ( 5, 1, 1 ) R3 ( 2, 4, 1 ) R4 ( 4, 2, 1 ) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Effectiveness Evaluation The security policy used for validating the effectiveness of the framework is shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Since any effective interaction is bidirectional in SDN networks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', the subject’s host and the object’s host must be ensured they can access each other in the data plane, so that we design the security policy as 4 access control relationships (R1 ∼ R4) to ensure h1 (1) and h5 (5) can access each other, h2 (2) and h4 (4) can access each other, and all the relationships of SPM are set as fixed=1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', having been updated by user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In the following, the effectiveness evaluations towards the framework will be carried out under 4 different scenarios at runtime, they are the effectiveness after loading the flow entries, the effectiveness after cutting down the path and the effectiveness after changing SPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Effectiveness after Loading the Flow Entries The purpose of this experiment is to validate whether the data plane after loading the generated flow entries can hold the security properties defined by SPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' First of all, the subjects and objects of SPM shown in Table I, i,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', the 1, 2, 4 and 5, are transformed into their corresponding hosts in the data plane by using security policy transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Specifically, the 1 is transformed into 13 Figure 3: The topology of the virtual SDN network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' h1=( 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='1, sw1, 1 ), the 2 is transformed into h2=( 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='2, sw2, 1 ), the 4 is transformed into h4=( 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='4, sw9, 1 ) and the 5 is transformed into h5=( 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='5, sw10, 1 ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In the following, the path searching algorithm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', Algorithm 2, searches the shortest path between the subject’s host and the object’s host for each Ri ∈ SPM based on the latest topology generated from the topology file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' After that, the relationships shown in Table I have been transformed into 4 corresponding shortest connected paths (P1 ∼ P4) in the data plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' As shown in Figure 4, the R1 is transformed into P1={ h1, e1, e2, e3, h5 }, the R2 is transformed into P2={ h5, e3, e2, e1, h1 }, the R3 is transformed into P3={ h2, e4, e5, h4 } and the R4 is transformed into P4={ h4, e5, e4, h2 }, where the P1 and P2 are depicted with the blue lines, the P3 and P4 are depicted with the orange lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In the following, the flow entry generation module transforms each path into a set of flow entries deployed in the switches passed by the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Specifically, P1 and P2 are transformed into the flow entries deployed in the switches { sw1, sw5, sw8, sw10 }, while P3 and P4 are transformed into the flow entries deployed in the switches { sw2, sw6, sw9 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' After that, we execute the pingall instruction in the Mininet CLI and observe the reachability of the entire data plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' As shown in Figure 5, h1 and h5 can access each other, h2 and h4 can access each other either, so that the data plane after loading the generated flow entries has been proven that it can hold all the security properties defined by SPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 14 Figure 4: The shortest paths searched by Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Figure 5: The result of executing the pingall instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 15 1 e es e3pi@raspberrypi:/mininet/custom 口 X s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 *** Adding links: (hl, sl) (h2, s2) (h3, s7) (h4, s9) (h5, s10) (h6, sl1) (sl, s2) (sl, s3) (sl, s 5) (s2, s4) (s2, s6) (s3, s4) (s3, s7) (s4, s5) (s4, s7) (s4, s8) (s5, s6) (s5, s8) (s5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' s9) (s6, s9) (s7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' s8) (s7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' s10) (s8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' s9) (s8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' s10) (s8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' s11) (s9, s11) (s10, sl1) *** Configuring hosts hl h2 h3 h4 h5 h6 *** starting controller c0 *** starting ll switches s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 *** Starting CLI: mininet> pingall *** Ping: testing ping reachability hl > X X X h5 X h2 > x x h4 x x 13 XXXXX<- h4 -> x h2 X X n5 -> hl x x x X X X X X X <- 9u *** Results: 86% dropped (4/30 received) mininet> mininet>5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Effectiveness after Cutting down the Path The purpose of this experiment is to validate whether the security properties defined by SPM can be continuously hold in the data plane after the connected path between the subject’s host and the object’s host is shut down at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' If the framework can synchronously perceive this change from the topology and automatically find another new connected path to keep the data plane holding the security properties at runtime, then the framework will be proven to be effective under this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' First of all, we let h1 and h5 can access each other by loading the corresponding flow entries into the data plane, and the shortest path between h1 and h5 is initialed with { h1, e1, e2, e3, h5 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In the following, we make a continuous TCP traffic sent from h1 to h5 by using the iperf instruction in the Mininet CLI, we set the duration time of the experiment equals 60 seconds and record the throughput of the traffics in h5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' When the time reaches 23 seconds, we shut down the edge between sw8 and sw10 existed in the path P1 by using the instruction in the Mininet CLI and let the experiment going on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' When the experiment is finished, we read the data recorded in h5 and plot them in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' As shown in Figure 6, the throughput of the traffic sent from h1 to h5 is sharply declined after the edge is shut down at 23 seconds, and completely becomes zero from 25 seconds to 37 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' After 38 seconds, the traffic quickly returns to normal until the end of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The experimental result has illustrated that the framework can synchronously perceive the change caused by cutting down one edge between sw8 and sw10, and automatically find another new shortest connected path between h1 and h5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', { h1, e6, e7, e8, h5 }, so as to make the traffic returning to normal quickly and keep the data plane holding the security properties defined by SPM at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The new shortest connected path searched by Algorithm 2 after cutting down the using path is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Effectiveness after Changing SPM The purpose of this experiment is to validate whether the security properties defined by SPM can be continuously hold in the data plane after SPM is changed by the user at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' If the framework can synchronously perceive this change from SPM and automatically update the flow entries deployed in the switches to keep the data plane holding the evolved security properties at runtime, then the framework will be proven to be effective under this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' First of all, we let h1 and h5 can access each other by loading a corresponding flow entries into the data plane, then make a continuous TCP traffic sent from h1 to h5 by using the iperf instruction in the Mininet CLI, we set the duration time of the experiment equals 60 seconds and record all the throughput data of the traffic in h5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In the following, we first validate the effectiveness of the framework under the scenario of changing SPM from h1 and h5 can access each other to h2 and h5 can access each other at runtime, and the experimental result under this scenario is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' As shown in Figure 8, the throughput of the traffic sent from h1 to h5, which is depicted with the red line, is quickly declined when we load the new SPM into the controller at 29 seconds, and completely becomes 16 Figure 6: The blue line represents the throughput of the traffic sent from h1 to h5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The dotted line represents we shut down the connected path at 23 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Figure 7: The new connected path searched by Algorithm 2 at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 17 Throughput (Gbits/sec) the traffic from h1 to h5 shut down the path at runtime 20 60 Time (sec)e6 eA 1 e7 es es eszero after 31 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' From 31 seconds until to the end of the experiment, h5 can only receive the traffic sent from h2 which is depicted with the blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The experimental result illustrates the framework can synchronously perceive this change and keep the data plane holding the evolved security properties after changing SPM from h1 and h5 can access each other to h2 and h5 can access each other at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Figure 8: The red line represents the throughput of the traffic sent from h1 to h5, the blue line represents the throughput of the traffic sent from h2 to h5, the dotted line represents we change SPM at 29 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' After that, we further validate the effectiveness of the framework under the scenario of adding a new relationship into SPM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', adding h2 and h5 can access each other, at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The experimental result under this scenario is plotted in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' As shown in Figure 9, the throughput of the traffic sent from h1 to h5, which is depicted with the red line, still keeps normal before we load the new SPM into the controller at 43 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' From 44 seconds until to the end of the experiment, h5 can receive the continuous traffic sent from h2 which is depicted with the blue line, and can also receive the traffic sent from h1 at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Due to the crowding of the traffic sent from h2, the throughput of the traffic from h1 is declined from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='71GB/s to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='99 GB/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The throughput of the traffic sent from h2 is still kept between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='6GB/s and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='2GB/s after 44 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The experimental result illustrates the framework can synchronously perceive this change and keep the data plane holding the evolved security properties after adding a new relationship into SPM at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 18 Throughput (Gbits/sec) the traffic from h1 to h5 the traffic from h2 to h5 load the new SPM at runtime Time (sec)Figure 9: The red line represents the throughput of the traffic sent from h1 to h5, the blue line represents the throughput of the traffic sent from h2 to h5, the dotted line represents we change SPM at 43 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Performance Evaluation As the critical algorithm used for implementing the security policy trans- formation, the performance of the path searching algorithm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', Algorithm 2, needs to be further evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' First of all, the sum of access control relation- ships of the security policy (SPM) is denoted as M, and the sum of OpenFlow switches in the topology is denoted as N in this performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Then by leveraging the Python programming, the execution time of Algorithm 2 have been recorded in milliseconds (ms) for calculating the shortest paths under set- ting the different value of M and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The experimental result is plotted in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' As shown in Figure 10, with gradually amplifying the value of M from 2 to 10, and the value of N from 11 to 400 respectively, the execution time of Algorithm 2 shows an obvious exponential upward trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Moreover, accord- ing to the description of Algorithm 2, the time complexity for calculating only one shortest path will reach O(N 2), because the Algorithm 2 needs to create a dynamic matrix djk[N][N] and further calculates the while loop, so that the time complexity for transforming all the access control relationships defined by SPM into their corresponding shortest paths will reach O(M × N 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Conclusion In this paper, we propose a practical runtime security policy transforma- tion framework for SDN networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' First of all,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' we specify the security policies used by SDN networks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' such as access control policies or firewall policies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' as a 19 Throughput (Gbits/sec) the traffic from h1 to h5 一the traffic from h2 to h5 load the new SPM at runtime Time (sec)Figure 10: The execution time of Algorithm 2 recorded in milliseconds (ms) for calculating the shortest paths under setting the different value of M and N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' where M represents the sum of access control relationships of SPM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' N represents the sum of OpenFlow switches in the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' system model of security policy (SPM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' SPM is of a high-level system model without containing any underlying information of data plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' From the theoret- ical level, we establish the system model for SDN network and propose a formal method to transform SPM into the corresponding flow entries automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The flow entry transformed from SPM is of a low-level system model containing the underlying information of data plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' From the practical level, we propose a runtime security policy transformation framework which consists of the se- curity policy module, topology discovery module, runtime monitoring module, path generation module, as well as flow entry generation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Leveraging these functional modules, the framework can solve the problem of how to find a connected path for each relationship defined by SPM in the data plane, how to transform the path into the system model of flow entries, as well as how to generate the practical flow entries by using the system model of flow entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' In order to validate the feasibility and effectiveness of the framework, we set up an experimental system and implement the framework by using POX con- troller and Mininet emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The experimental result illustrate the framework is completely effective at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' However, there still exists some problems needed to be further researched in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' The current path searching algorithm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=', Algorithm 2, used by the framework is improved from the classic Dijkstra algorithm and finds the shortest path by calculating the minimum number of hops in the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' in the real SDN networks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' the problem of searching a connected path between the two hosts need to consider the matters of quality of service (QoS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' 20 350 when M =2 when M =4 M 300- when M =6 when M =8 when M =10 250 (sw) e 200 Performance W 150 P 100 - 50 - +0 0 50 100 150 200 250 300 350 400 Sum of OpenFlow Switches (N)load balance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' as well as some specific requirements about the traffic engineering at runtime,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' so that the framework needs to be further improved by employing some novel path searching algorithms based on multi-object optimization[29] or reinforcement learning[30] methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' Acknowledgment This paper has been sponsored and supported by National Key Research and Development Program of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='2018YFB0803400), Doctoral Foundation of Qingdao Binhai University (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content=' BS2022A10), partially supported by Key Program of National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQfSQc5/content/2301.03790v1.pdf'} +page_content='61932013).' metadata={'source': 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With this data and additional +information on the person’s background, one can easily infer an individual’s private data. Many differentially private iterative +algorithms have been proposed in interactive settings to protect an individual’s privacy from these inference attacks. The +existing approaches adapt the method to compute differentially private(DP) centroids by iterative Llyod’s algorithm and +perturbing the centroid with various DP mechanisms. These DP mechanisms do not guarantee convergence of differentially +private iterative algorithms and degrade the quality of the cluster. Thus, in this work, we further extend the previous work on +‘Differentially Private 𝑘-Means Clustering With Convergence Guarantee’ by taking it as our baseline. The novelty of our +approach is to sub-cluster the clusters and then select the centroid which has a higher probability of moving in the direction +of the future centroid. At every Lloyd’s step, the centroids are injected with the noise using the exponential DP mechanism. +The results of the experiments indicate that our approach outperforms the current state-of-the-art method, i.e., the baseline +algorithm, in terms of clustering quality while maintaining the same differential privacy requirements. The clustering quality +significantly improved by 4.13 and 2.83 times than baseline for the Wine and Breast_Cancer dataset, respectively. +Keywords +differential privacy, 𝑘-means clustering, convergence guarantee +1. Introduction +Achieving extraordinary results is dependent on the data +on which the machine learning models are trained. Data +curators have a responsibility to provide datasets such +that the privacy of data is not compromised. However, +attackers use other public datasets to perform inference +and adversarial attacks to get information about an indi- +vidual in the dataset. Differential privacy is a potential +technique for giving customers a mathematical guarantee +of the privacy of their data[1]. There are two fundamen- +tal settings in which differential privacy is used on data: +in interactive setting data curator holds the data and re- +turns the response based on the queries requested by +third parties; while in non-interactive setting the curator +sanitized the data before publishing[2]. +Iterative clustering algorithms provide important in- +sights about the dataset, which helps in a large number of +applications. They are prone to privacy threats because +they can reveal information about an individual with ad- +ditional knowledge. Existing approaches obtain the set +of centroids using Lloyd’s K-means algorithm, then per- +turb them with a differentially private mechanism to add +CIKM-PAS’22: PRIVACY ALGORITHMS IN SYSTEMS (PAS) Workshop, +Conference on Information and Knowledge Management, October 21, +2022, CIKM-PAS +*Corresponding author. +†These authors contributed equally. +� devvrat.joshi@iitgn.ac.in (D. Joshi); janvi.thakkar@iitgn.ac.in +(J. Thakkar) +© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License +Attribution 4.0 International (CC BY 4.0). +CEUR +Workshop +Proceedings +http://ceur-ws.org +ISSN 1613-0073 +CEUR Workshop Proceedings (CEUR-WS.org) +privacy [3]. In contrast to Lloyd’s K-means algorithm, +which guarantees convergence, these algorithms do not +provide any convergence guarantee. Getting differen- +tially private centroids might not help in getting quality +inferences because of this non-convergence. We studied +an existing approach that provides this guarantee and +converges in twice the number of iterations to Lloyd’s al- +gorithm while maintaining the same differential privacy +requirements as existing works [4] [5]. Their algorithm +perturbs the centroids in a random direction from the +center of the cluster. However, this lowers the quality of +clustering, which is necessary for making inferences. +In this work, we propose a variant of the existing ap- +proach, which provides better clustering quality while +using the same privacy budget. We used the intuition +of Lloyd’s algorithm that the next centroid will move +in the direction where there is a higher number of data +points. Finally, we give the mathematical proof that our +approach at any instance gives better clustering quality +than the existing approaches. We have tested our ap- +proach on breat_cancer, wine, iris, and digits datasets. +We were able to get a significant improvement from the +previous approach in terms of clustering quality. +Interactive setting implies that the dataset is not dis- +closed to the user, however, the data curator returns the +response of each query received from the user by manip- +ulating it using DP strategy. +Our main contribution includes: +1. We proposed SubClustering approach which has +better clustering quality than the baseline (which +is the current SOTA in terms of clustering qual- +arXiv:2301.02896v1 [cs.LG] 7 Jan 2023 + +ity). For the Wine and Breast_cancer dataset, the +clustering quality improved by 4.13 and 2.83 times +respectively. +2. In addition to improving the clustering quality, +our algorithm used same privacy budget as that +of the existing work. +2. Related Work +The concept of differential privacy has inspired a plethora +of studies, particularly in the area of differentially private +k-means clustering [6][7][8] in an interactive setting. The +important mechanisms of DP in the literature include: +the Laplace mechanisms (LapDP) [9], the exponential +mechanisms (ExpDP) [10], and the sample and aggregate +framework [11]. To achieve differential privacy, many im- +plementations included infusing Laplace noise into each +iteration of Lloyd’s algorithm. The proportion of noise +added was based on a fixed privacy budget. Some of the +strategies for allocating privacy budget included splitting +the overall privacy budget uniformly to each iteration +[12]. However, this requires us to calculate the number of +iterations for the convergence, prior to the execution of +algorithm, thus increasing the computational cost. Fur- +ther, researchers overcome this weakness by allocating +theoretically guaranteed optimal allocation method [6], +but the major assumption taken in this approach was +that every cluster has the same size, which does not align +with the real-world datasets. In another work, Mohan +et al. [8] proposed GUPT, which uses Lloyd’s algorithm +for local clustering of each bucket where the items were +uniformly sampled to different buckets. The final result +was the mean of locally sampled points in each bucket +with added Laplace noise. But, the clustering quality of +GUPT was unsatisfying because a large amount of noise +was added in the aggregation stage. +Based on the study of past literature on differentially +private k-means clustering, Zhigang et al. [3] concluded +that convergence of an iterative algorithm is important to +the clustering quality. To solve this, they introduced the +concept of the convergent zone and orientation controller. +With the help of a convergent zone and orientation con- +troller, they further create a sampling zone for selecting +a potential centroid for the 𝑖𝑡ℎ iteration. The approach +iteratively adds noise with an exponential mechanism +(ExpDP) by using prior and future knowledge of the po- +tential centroid at every step of Lloyd’s algorithm. The ap- +proach maintains the same DP requirements as existing +literature, with guaranteed convergence and improve- +ment in clustering quality. However, their algorithm +perturbs the centroids in a random direction from the +center of the cluster, degrading the quality of clustering. +Thus, in this work, we further build upon the approach +and significantly improve the clustering quality with the +same epsilon privacy. +3. Preliminaries +The definitions used in this work are briefly discussed +in this section. The following is a formal definition of +Differential Privacy: +Definition 1 (𝜖-DP [9]). A randomised mechanism T +is 𝜖- differentially private if for all neighbouring datasets +𝑋 and 𝑋′ and for an arbitrary answer 𝑠 ∈ 𝑅𝑎𝑛𝑔𝑒(𝑇), T +satisfies +𝑃𝑟[𝑇(𝑋) = 𝑠] ≤ 𝑒𝑥𝑝(𝜖) · 𝑃𝑟[𝑇(𝑋′) = 𝑠], +where 𝜖 is the privacy budget. +Here, 𝑋 and 𝑋′ differ by only one item. Smaller val- +ues of 𝜖 imply a better privacy guarantee. It is because +the difference between the two neighboring datasets is +reflected by the privacy budget. In this work, we use the +ExpDP and LapDP. In exponential DP for non-numeric +computation, they introduce the concept of scoring func- +tion 𝑞(𝑋, 𝑥), which represents the effectiveness of the +pair (𝑋, 𝑥). Here 𝑋 is the dataset and 𝑥 is the response +to the 𝑞(𝑋, 𝑥) on X. +The formal definition of Exponential DP mechanism +is defined as follow: +Definition +2 +(Exponential +Mechanism +[10]). +Given a scoring function of a dataset 𝑋, 𝑞(𝑋, 𝑥), +which reflects the quality of query respond x. +The +exponential mechanism T provides 𝜖-differential privacy, +if 𝑇(𝑋) = {𝑃𝑟[𝑥] ∝ 𝑒𝑥𝑝( 𝜖·𝑞(𝑋,𝑥) +2Δ𝑞 +)}, where ∆𝑞 is the +sensitivity of scoring function q(X,x), 𝜖 is the privacy +budget. +Definition 3 (Convergent & Sampling Zones[3]). +A region whose points satisfies the condition: { Node S: +‖𝑆 − 𝑆𝑖 +(𝑡)‖ < ‖𝑆𝑖 +(𝑡−1) − 𝑆𝑖 +(𝑡)‖} is the convergent zone. +𝑆𝑖 +(𝑡) is defined as the mean of 𝐶𝑖 +(𝑡). A sub-region inside +convergent zone is defined as a sampling zone. +Definition 4 (Orientation Controller[3]). 𝑋𝑖 +(𝑡) is +a direction from the center of the convergent zone to a point +on its circumference. This is the direction along which the +center of the sampling zone will be sampled, defined as the +orientation controller. +4. Approach +In this section, we explain our proposed approach and +the baseline approach. +4.1. Overview - KMeans Guarantee +(Baseline) +We took "Differentially Private K-Means Clustering with +Convergence Guarantee" [3] as our baseline and im- +proved the clustering quality by further building on it. + +Figure 1: Overview of KMeans Guarantee Approach +The key concept of the algorithm is to use ExpDP to in- +troduce bounded noise into centroids at each iteration of +Lloyd’s algorithm. The technique is designed in a way +that it ensures the new centroid is different from the cen- +troid of Lloyd’s algorithm while maintaining constraint +given in Lemma 1. The constraint guarantees that the +perturbed centroid will eventually converge with the +centroid of Lloyd’s algorithm. +Their algorithm has four main steps to update the +centroids at each Lloyd step t [3]. The overview of their +approach can be seen in (Figure : 1). +1. Let the differentially private centroid at iteration +𝑡−1 for a cluster 𝑖 be 𝑆𝑖ˆ (𝑡−1). Using this centroid, +run one iteration of Lloyd’s algorithm to get the +current Lloyd’s centroid 𝑆𝑖 +(𝑡) for each cluster 𝑖. +2. Using 𝑆𝑖 +(𝑡) and 𝑆𝑖 +(𝑡−1), generate a conver- +gent zone for each cluster 𝑖 as described in +𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 3. +3. Generate a sampling zone in the convergence zone +and an orientation controller 𝑋𝑖 +(𝑡) for each cluster +i as defined in 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 3 𝑎𝑛𝑑 4 respectively. +4. Sample a differentially private 𝑆𝑖ˆ (𝑡) with ExpDP +in the sampling zone generated in step 3. +The definition for the convergent zone (for convergence +guarantee) and sampling zone (for centroid updating) is +defined in Definition 3. +4.2. Overview - SubCluster Guarantee +We build upon the KMeans Guarantee algorithm to +achieve better clustering quality. Our idea differs from +the baseline in terms of creating a sampling zone. For +each cluster, we execute Lloyd’s algorithm over its con- +vergent zone to generate its sub-clustering. Further, we +assign each sub-cluster with a probability linearly pro- +portional to the number of points it contains. Finally, we +sample the sub-cluster based on the assigned probability +and define it as the sampling zone of the convergent zone. +Drawing analogy from the KMeans Guarantee algorithm, +our orientation controller is this sub-clustering and sam- +pling technique. Intuitively, our algorithm ensures that +Algorithm 1: Differentially Private 𝑘−Means +SubClustering Algorithm +Input: X = {𝑥1, 𝑥2, ...., 𝑥𝑁}: Dataset with N +data points +k: number of clusters +𝜖𝑒𝑥𝑝: ExpDP privacy budget +𝜖𝑙𝑎𝑝: Laplacian privacy budget for the converged +centroids. +𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙𝐾: number of sub-clusters per cluster +Output: S: Final clustering centroids +1 Select 𝑘 centroids S(0) = (𝑆(0) +1 , 𝑆(0) +2 , ..., 𝑆(0) +𝑘 ) +uniformly from X. +2 𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝐹𝑜𝑟𝐿𝑙𝑜𝑦𝑑 = number of iterations to +run the algorithm. +3 for iters i in 𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝐹𝑜𝑟𝐿𝑙𝑜𝑦𝑑 do +4 +for each Cluster i at Iteration t do +5 +𝐶(𝑡) +𝑖 +← assign each 𝑥𝑗 to its closest +centroid 𝑆𝑖 +𝑡−1; +6 +𝑆𝑖 +𝑡 ← centroid of 𝐶𝑖 +𝑡; +7 +𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒𝑖 +(𝑡) ← List of data +points inside the spherical region having +𝑆𝑖 +𝑡 and 𝑆𝑖 +𝑡−1 as the endpoints of its +radius. +8 +𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑖 +(𝑡) ← run Algorithm 2 +using 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒𝑖 +(𝑡) , +𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙𝐾; +9 +𝑆𝑖ˆ (𝑡) ← sample from 𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑖 +(𝑡) +using ExpDP with 𝑞 and 𝜖𝑒𝑥𝑝; +10 +𝑆𝑖 +(𝑡) ← 𝑆𝑖ˆ (𝑡) +11 Publish: 𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑖 +(𝑡), 𝑞, 𝜖𝑒𝑥𝑝, 𝑆𝑖 +(𝑡) +12 S ← add laplace noise with 𝜖𝑙𝑎𝑝 to S(𝑡); +Algorithm 2: SubClusterSamplingAlgorithm +Input: ConvergentZone: Convergent Zone +internalK: Subclustering K +Output: 𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑡 +𝑖 +1 S(𝑡): Mean of 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒𝑖 +(𝑡) +2 ConvergentZoneClusters ← Cluster +ConvergentZone using Lloyd’s algorithm and +𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙𝐾 +3 ConvergentZoneProbability ← Assign +probabilities to the +ConvergentZoneClusters proportional to +the number of points inside each cluster. +4 SamplingZonei +(t) ← Sample a cluster from +the ConvergentZoneClusters using +ConvergentZoneProbability +5 Return: SamplingZonei +(t); + +Cluster i at Iteration t ( +415 +sampling zone +(）人 +α(t+1) +convergence zone +orientation +controllerFigure 2: Overview of SubCluster Guarantee Approach +the sampling zone lies towards the region containing a +higher number of data points in an expected case. With +this, we guarantee that our differentially private centroid +moves in the direction where the number of data points +is higher, incorporating the intuition of Lloyd’s algorithm +without compromising on the 𝜖-differential privacy. The +probability of a differentially private centroid at 𝑖 − 1𝑡ℎ +iteration to move in the direction of a more populated re- +gion at the 𝑖𝑡ℎ step of Lloyd’s algorithm is also high. Thus, +we introduce the concept of sub-clustering in the conver- +gent zone and consequently sample one sub-cluster as +our sampling zone. +We sample the centroid from the sampling zone using +the ExpDP mechanism. Finally, we inject Laplace noise +in the centroids of the clustering when our algorithm +converges. It is because the differentially private cen- +troids obtained are a subset of one of the local minima +at which Lloyd’s algorithm converges. The overview of +the proposed approach can be seen in (Figure : 2). We +show that a randomized iterative algorithm satisfies an +invariant (given in the claim of Lemma 1) and always +converges (Proof: refer Lemma 1). Finally, we show +that the SubCluster algorithm is a randomized iterative +algorithm that satisfies the invariant(given in Lemma 1) +(Proof: Refer Lemma 2). +We have four main steps to update the centroids at +each Lloyd step t. +1. Let the differentially private centroid at iteration +𝑡−1 for a cluster 𝑖 be 𝑆𝑖ˆ (𝑡−1). Using this centroid, +run one iteration of Lloyd’s algorithm to get the +current Lloyd’s centroid 𝑆𝑖 +(𝑡) for each cluster 𝑖. +2. Using 𝑆𝑖 +(𝑡) and 𝑆𝑖 +(𝑡−1), generate a conver- +gent zone for each cluster 𝑖 as described in +𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 3. +3. SubCluster the convergence zone and sample one +of the sub-cluster as our sampling zone based on +the probability assigned to each sub-cluster. The +probability assignment is directly proportional to +the number of points in each sub-cluster. +4. Sample a differentially private 𝑆𝑖ˆ (𝑡) with EXpDP +in the sampling zone generated in step 3. +Our approach surpasses the baseline approach in terms +of clustering quality while maintaining the same DP re- +quirements as that of the KMeans Guarantee approach, +which is evident from the results obtained (Figure : +3). The better clustering quality is a result of our sub- +clustering strategy to perturb centroid with a higher prob- +ability than the baseline approach towards the direction +of the actual centroid generated by Lloyd’s algorithm. +The pseudo-code of our approach is shown in the Algo- +rithm 1 and Algorithm 2. +Lemma 1: [3] A randomised iterative algorithm +𝜏 is convergent if, in 𝐶(𝑡) +𝑖 (Cluster i at iteration t), +𝑆𝑖ˆ (𝑡)(sampled centroid using 𝜏), 𝑆𝑖 +(𝑡−1)(centroid before +recentering) and 𝑆𝑖 +(𝑡)(centroid of 𝐶(𝑡) +𝑖 ) satisfies the in- +variant, ||𝑆𝑖ˆ (𝑡)−𝑆𝑖 +(𝑡)|| < ||𝑆𝑖 +(𝑡)−𝑆𝑖 +(𝑡−1)|| in Euclidean +distance, ∀𝑡, 𝑖. +We reproduce this lemma from our baseline approach +[3]. Lemma1 and Lemma 2 together provides the com- +pleteness and proof for the convergence of our approach. +If the distance between the sampled centroid 𝑆ˆ(𝑡) +𝑖 +from +the 𝐶(𝑡) +𝑖 +and the new centroid 𝑆(𝑡) +𝑖 +is less than the dis- +tance between the new 𝑆(𝑡) +𝑖 +and the old centroid 𝑆(𝑡−1) +𝑖 +, +then the random iterative algorithm will always converge. +Intuitively, the loss of 𝐶(𝑡) +𝑖 +is minimum if the mean of +𝐶(𝑡) +𝑖 +is taken as centroid. But, if we slightly shift from +the mean of 𝐶(𝑡) +𝑖 , then the loss will increase. However, if +we can ensure that any sampled point from 𝐶(𝑡) +𝑖 +fulfills +the condition: ||𝑆𝑖ˆ (𝑡) − 𝑆𝑖 +(𝑡)|| < ||𝑆𝑖 +(𝑡) − 𝑆𝑖 +(𝑡−1)||, it +will lead to a lesser loss than 𝐽𝑆(𝑡−1) +𝑖 +, thus, resulting into +convergence of the randomised iterative algorithm. For +the mathematical proof, refer [3]. +Lemma 2: Differentially Private 𝑘−Means SubClus- +tering approach (SubClustering) is a randomised iterative +algorithm that satisfies the invariant ||𝑆𝑖ˆ (𝑡) − 𝑆𝑖 +(𝑡)|| < +||𝑆𝑖 +(𝑡) − 𝑆𝑖 +(𝑡−1)||. +Proof: SubClustering is an iterative algorithm that +samples a set of centroids for each iteration with Ex- +pDP mechanism, thus, making it a randomised itera- +tive algorithm. It subclusters the points lying inside +𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡) +𝑖 . After subclustering, it samples +one subcluster (sampling zone) with the assigned proba- +bilities (linearly proportional to the number of data points +in subcluster). Finally, it samples a datapoint from the +sampled subcluster with ExpDP and call it as the cen- + +Cluster i at iteration t (C(t) +subclusters +S;(t-1) > S;(t-1) +S,(t+7) +Srt+s +convergent zone +≤.(t) +sampling zoneFigure 3: Above figures plots the graph between costGap and epsilon budget for two approaches, the baseline as KmeansGuar- +antee and our approach SubClusterGuarantee. The algorithm was tested on four dataset, Digits (top-left), Wine (top-right), +Breast Cancer (bottom-left), and Iris (bottom-right) datasets. +troid of 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡) +𝑖 . Thus, our sampling zone +always lies inside 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡) +𝑖 . Therefore, the +sampled point lies inside 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡) +𝑖 +and it sat- +isfies the invariant ||𝑆𝑖ˆ (𝑡) − 𝑆𝑖 +(𝑡)|| < ||𝑆𝑖 +(𝑡) − 𝑆𝑖 +(𝑡−1)||. +5. Experimental Setup +5.1. Dataset Used +We used following four datasets to test our work Sub- +Cluster Guarantee upon the baseline: +1. Iris [13] dataset comprises total of 150 datapoints +with four features and three classes. +2. Wine[13] dataset comprises total of 178 data- +points with 13 features and three classes. +3. Breast Cancer[13] dataset comprises total of +569 datapoints with 30 features and two classes. +4. Digits[13] dataset comprises of 1797 datapoints +with 64 dimensions and 10 classes. +5.2. Metric for Clustering Quality +To evaluate the clustering quality, we used the following +equation to calculate the normalised difference between +the differentially private algorithms (here, SubCluster +Guarantee approach) (𝐶𝑜𝑠𝑡𝐷𝑃 ) and Lloyd’s algorithm +(𝐶𝑜𝑠𝑡𝐿𝑙𝑜𝑦𝑑): +𝐶𝑜𝑠𝑡𝐺𝑎𝑝 = |𝐶𝑜𝑠𝑡𝐷𝑃 − 𝐶𝑜𝑠𝑡𝐿𝑙𝑜𝑦𝑑| +𝐶𝑜𝑠𝑡𝐿𝑙𝑜𝑦𝑑 +(1) +The smaller CostGap [3] represents the better quality of +clustering. In the experiments, we compare the clustering +quality of SubCluster Guarantee with KMeans Guarantee. +6. Results and Discussion +We tested our algorithm on four datasets. All the datasets +have different dimensions ranging from 4 to 64 dimen- +sions and training sets ranging from 150 to 1800. As +defined in metric smaller gap represents the better clus- +tering quality. From the (Figure : 3) we can observe +that, cost gap for all the dataset is smaller or equal to +the baseline. Thus, it is evident that our algorithm has +better clustering quality than the existing work for all the +datasets experimented. We varied internalK (parameter +for number of sub-clusters) from 2 to 5. +Each experiment was conducted 30 times in the case +of the Iris, Wine, and Breast cancer dataset and 10 times +for digits dataset due to computational constraints. Fi- +nally, for each dataset, we took the average of all the +experiments as our final result for plotting the graphs. + +KmeansGuaranteevsSubClusterGuaranteeDataset:Digits +KmeansGuaranteevsSubClusterGuaranteeDataset:Wine +0.200 +0.08 +KmeansGuarantee +KmeansGuarantee +0.175 +SubClusterGuarantee +0.07 +SubClusterGuarantee +0.150 +0.06 +0.125 +0.05 +costGap +0.04 +0.075 +E00 +0.050 +0.02 +0.025 +0.01 +0.000 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +epsilon +epsilon +KmeansGuaranteevsSubClusterGuaranteeDataset:BreastCancer +KmeansGuaranteevsSubClusterGuaranteeDataset:Iris +0.035 +14 +KmeansGuarantee +12 +SubClusterGuarantee +0.030 +10 +0.025 +KmeansGuarantee +8 +SubClusterGuarantee +6 +0.015 +4 +0.010 +2 +0.005 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +epsilon +epsilonFigure 4: Above figures plots the graph between costGap and epsilon budget for different internalK in SubClusterGuarantee +Algorithm. The algorithm was tested for internalK=2,3,4,5 for all the four datasets, Digits (top-left), Wine (top-right), Breast +Cancer (bottom-left), and Iris (bottom-right). Please note: K and internalK are the same parameter +Comparing the SubCluster Guarantee (proposed ap- +proach) and K-means Guarantee approach (baseline) by +taking an average of all the cost gaps for varied epsilon, +and finally taking the ratio between K-means and Sub- +Cluster approach: +1. In case of Iris dataset, the cost gap is 1.1 times +smaller than baseline algorithm. +2. In case of Wine dataset, the cost gap is 4.13 times +smaller than baseline algorithm. +3. In case of Breast_Cancer dataset, the cost gap +is 2.83 times smaller than baseline algorithm. +4. In case of Digits dataset, the cost gap is almost +same as that of baseline algorithm. +6.1. Detailed Analysis +1. Iris: Iris dataset has four dimensions and a very +small training set of 150 data points. Our al- +gorithm achieves better clustering quality than +the baseline algorithm for smaller epsilon values. +Since the number of data points is less in Iris, the +impact of sub-clustering reduces, resulting in its +performance similar to that of the baseline ap- +proach. From (Figure : 4), we can observe that +changing the value of intenalK has a small impact +on the costGap due to a small number of points +in each sub-cluster. This is because there is a pos- +sibility that a sub-cluster has no data point when +internalK is increased causing zero probability +sub-cluster regions. +2. Wine: The wine dataset has 13 dimensions and +178 data points in the training set. Our algorithm +performs significantly better than the baseline, as +observed in (Figure : 3). It is because the baseline +algorithm is constrained to choose a theta in any +abrupt direction ranging from [−𝜋/2, 𝜋/2] as +shown in (Figure : 1). In contrast, our algorithm +shifts the centroids in the direction where the fu- +ture centroid of Lloyd’s algorithm is more likely +to move (in the expected case). From (Figure : 4), +it is evident that internalK=4 for the wine dataset +performs better than the rest of the internalK val- +ues. Here, the number of dimensions is more than +Iris. Therefore, the spatial arrangement will be in +an n-sphere which allows better sub-clustering. +3. Breast_Cancer: Breast_Cancer dataset has 569 +data points in its training set and 30 dimensions. +Our algorithm performs exceptionally better than +the baseline, with internalK equal to 4. From +(Figure : 3), we can observe that there is no +monotonous trend for the costGap. Trends are +visible in other datasets due to the larger num- +ber of classification classes, whereas this dataset +has only two classes. Thus, adding Laplace noise +does not have a relation to the clustering quality. +Increasing the internalK improves the clustering + +VaryinginternalKforSubClusteringDataset:Digits +VaryinginternalKforSubClusteringDataset:Wine +K=2 +0.0035 +K=2 +0.20 +K=3 +K=3 +K=4 +K=4 +0.0030 +K=5 +K=5 +0.15 +0.0025 +0.10 +0.0020 +0.05 +0.0015 +0.00 +0.0010 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +epsilon +epsilon +VaryinginternalkforSubClusteringDataset:BreastCancer +VaryinginternalKforSubClusteringDataset:Iris +K=2 +10 +K=2 +0.025 +K=3 +K=3 +K=4 +K=4 +0.020 +8 +K=5 +h +0.010 +0.005 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +epsilon +epsilonquality, with internalK being 4 having the least +loss. It is because this dataset has a high number +of dimensions and a larger number of training +points than other datasets. +4. Digits: It has 64 dimensions and 1797 data points +in the training dataset. Although it has a large +number of dimensions, our algorithm has a very +small improvement over the baseline algorithm as +seen in (Figure : 3). Because of the higher time +complexity of our algorithm, it is hard to tune +the internalK parameter. As the number of sam- +ples in a dataset increases, the internalK should +increase because a single cluster can contain a +large number of data points. But, due to limited +computational resources, we were not able to ex- +periment with it further. We took internalK to +be 5 for our experiments as it performed best in +the range [2, 5] as in the (Figure : 4). One of the +intriguing findings in the dataset’s results is that +the curves based on the internalK have a clearly +evident trend, which is a result of the large num- +ber of training data points. +Our proposed algorithm significantly improves over the +baseline in terms of clustering quality, especially for the +wine and breast cancer dataset. In addition our algorithm +maintains the same DP requirements as that of existing +works. +7. Conclusion +This work presents a novel method for improving the +clustering quality of differentially private k-means al- +gorithms while ensuring convergence. The novelty of +our approach is the sub-clustering of the cluster to select +the differentially private centroid, which has a higher +probability of moving in the direction of the next cen- +troid. We proved that our work surpasses the current +state-of-the-art algorithms in terms of clustering quality. +Especially for the Wine and Breast_Cancer dataset, the +clustering quality was significantly improved by 4.13 and +2.83 times than the baseline. In addition, we maintain +the same DP requirements as that of baseline and other +existing approaches. +8. Future Work +• In this work, we proved our claim using empirical +results. We further plan to validate the results +by providing mathematical bounds for the con- +vergence degree and rate of the SubClustering +Lloyd’s algorithm. In terms of clustering qual- +ity, the proposed algorithm in this work is com- +pared with k-means guarantee clustering only; +to prove the effectiveness of our work, we plan +to experiment with other algorithms in the lit- +erature including, PrivGene [14], GUPT [8] and +DWork [7]. +• The DP requirements in this work are the same +as that of past literature, but in the future, we +plan to explore ways to improve the current DP +guarantees while maintaining the same clustering +quality as in this work. +• We used Exponential and Laplace mechanisms +of DP in the proposed approach; we further plan +to explore the third mechanisms, i.e., sample and +aggregate framework, by integrating it with the +current algorithm. +• In our algorithm, the number of data points inside +a cluster is variable. Thus we plan to choose an +internalK, custom to the size of the cluster to +improve the clustering quality. +Acknowledgement +We would like to thank Prof. Anirban Dasgupta +(IIT Gandhinagar) for his continuous support and +guidance throughout the research. +References +[1] C. Dwork, Differential privacy: A survey +of results, in: International conference on +theory and applications of models of com- +putation, Springer, 2008, pp. 1–19. +[2] A. Narayanan, Data privacy: The non- +interactive setting, The University of Texas +at Austin, 2009. +[3] Z. Lu, H. Shen, +Differentially private k- +means clustering with convergence guar- +antee, IEEE Transactions on Dependable +and Secure Computing (2020). +[4] D. Su, J. Cao, N. Li, E. Bertino, H. Jin, Dif- +ferentially private k-means clustering, in: +Proceedings of the sixth ACM conference +on data and application security and pri- +vacy, 2016, pp. 26–37. +[5] J. Lei, Differentially private m-estimators, +Advances in Neural Information Processing +Systems 24 (2011). +[6] D. Su, J. Cao, N. Li, E. Bertino, M. Lyu, H. Jin, +Differentially private k-means clustering +and a hybrid approach to private optimiza- +tion, ACM Transactions on Privacy and +Security (TOPS) 20 (2017) 1–33. +[7] C. Dwork, A firm foundation for private +data analysis, Communications of the ACM +54 (2011) 86–95. + +[8] P. Mohan, A. Thakurta, E. Shi, D. Song, +D. Culler, Gupt: privacy preserving data +analysis made easy, in: Proceedings of the +2012 ACM SIGMOD International Confer- +ence on Management of Data, 2012, pp. 349– +360. +[9] C. Dwork, F. McSherry, K. Nissim, A. Smith, +Calibrating noise to sensitivity in private +data analysis, in: Theory of cryptography +conference, Springer, 2006, pp. 265–284. +[10] F. McSherry, K. Talwar, Mechanism design +via differential privacy, in: 48th Annual +IEEE Symposium on Foundations of Com- +puter Science (FOCS’07), IEEE, 2007, pp. 94– +103. +[11] K. Nissim, S. Raskhodnikova, A. Smith, +Smooth sensitivity and sampling in private +data analysis, in: Proceedings of the thirty- +ninth annual ACM symposium on Theory +of computing, 2007, pp. 75–84. +[12] A. Blum, C. Dwork, F. McSherry, K. Nis- +sim, Practical privacy: the sulq framework, +in: Proceedings of the twenty-fourth ACM +SIGMOD-SIGACT-SIGART symposium on +Principles of database systems, 2005, pp. +128–138. +[13] A. Asuncion, Uci machine learning reposi- +tory, university of california, irvine, school +of information and computer sciences, +http://www. ics. uci. edu/˜ mlearn/MLRepos- +itory. html (2007). +[14] J. Zhang, X. Xiao, Y. Yang, Z. Zhang, +M. Winslett, Privgene: differentially pri- +vate model fitting using genetic algorithms, +in: Proceedings of the 2013 ACM SIGMOD +International Conference on Management +of Data, 2013, pp. 665–676. + diff --git a/F9E1T4oBgHgl3EQfFAMn/content/tmp_files/load_file.txt b/F9E1T4oBgHgl3EQfFAMn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b30d2edd605b8e3186d04306b4b1c7e9cc3263dd --- /dev/null +++ b/F9E1T4oBgHgl3EQfFAMn/content/tmp_files/load_file.txt @@ -0,0 +1,433 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf,len=432 +page_content='𝑘-Means SubClustering: A Differentially Private Algorithm with Improved Clustering Quality Devvrat Joshi1,*,†, Janvi Thakkar1,*,† 1Indian Institute of Technology Gandhinagar, India Abstract In today’s data-driven world, the sensitivity of information has been a significant concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' With this data and additional information on the person’s background, one can easily infer an individual’s private data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Many differentially private iterative algorithms have been proposed in interactive settings to protect an individual’s privacy from these inference attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The existing approaches adapt the method to compute differentially private(DP) centroids by iterative Llyod’s algorithm and perturbing the centroid with various DP mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' These DP mechanisms do not guarantee convergence of differentially private iterative algorithms and degrade the quality of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Thus, in this work, we further extend the previous work on ‘Differentially Private 𝑘-Means Clustering With Convergence Guarantee’ by taking it as our baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The novelty of our approach is to sub-cluster the clusters and then select the centroid which has a higher probability of moving in the direction of the future centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' At every Lloyd’s step, the centroids are injected with the noise using the exponential DP mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The results of the experiments indicate that our approach outperforms the current state-of-the-art method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=', the baseline algorithm, in terms of clustering quality while maintaining the same differential privacy requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The clustering quality significantly improved by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='13 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='83 times than baseline for the Wine and Breast_Cancer dataset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Keywords differential privacy, 𝑘-means clustering, convergence guarantee 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Introduction Achieving extraordinary results is dependent on the data on which the machine learning models are trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Data curators have a responsibility to provide datasets such that the privacy of data is not compromised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' However, attackers use other public datasets to perform inference and adversarial attacks to get information about an indi- vidual in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Differential privacy is a potential technique for giving customers a mathematical guarantee of the privacy of their data[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' There are two fundamen- tal settings in which differential privacy is used on data: in interactive setting data curator holds the data and re- turns the response based on the queries requested by third parties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' while in non-interactive setting the curator sanitized the data before publishing[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Iterative clustering algorithms provide important in- sights about the dataset, which helps in a large number of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' They are prone to privacy threats because they can reveal information about an individual with ad- ditional knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Existing approaches obtain the set of centroids using Lloyd’s K-means algorithm, then per- turb them with a differentially private mechanism to add CIKM-PAS’22: PRIVACY ALGORITHMS IN SYSTEMS (PAS) Workshop, Conference on Information and Knowledge Management, October 21, 2022, CIKM-PAS Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' †These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' � devvrat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='joshi@iitgn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Joshi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' janvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='thakkar@iitgn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='in (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Thakkar) © 2022 Copyright for this paper by its authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Use permitted under Creative Commons License Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='0 International (CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' CEUR Workshop Proceedings http://ceur-ws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='org ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='org) privacy [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' In contrast to Lloyd’s K-means algorithm, which guarantees convergence, these algorithms do not provide any convergence guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Getting differen- tially private centroids might not help in getting quality inferences because of this non-convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' We studied an existing approach that provides this guarantee and converges in twice the number of iterations to Lloyd’s al- gorithm while maintaining the same differential privacy requirements as existing works [4] [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Their algorithm perturbs the centroids in a random direction from the center of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' However, this lowers the quality of clustering, which is necessary for making inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' In this work, we propose a variant of the existing ap- proach, which provides better clustering quality while using the same privacy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' We used the intuition of Lloyd’s algorithm that the next centroid will move in the direction where there is a higher number of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Finally, we give the mathematical proof that our approach at any instance gives better clustering quality than the existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' We have tested our ap- proach on breat_cancer, wine, iris, and digits datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' We were able to get a significant improvement from the previous approach in terms of clustering quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Interactive setting implies that the dataset is not dis- closed to the user, however, the data curator returns the response of each query received from the user by manip- ulating it using DP strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Our main contribution includes: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' We proposed SubClustering approach which has better clustering quality than the baseline (which is the current SOTA in terms of clustering qual- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='02896v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='LG] 7 Jan 2023 ity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' For the Wine and Breast_cancer dataset, the clustering quality improved by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='13 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='83 times respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' In addition to improving the clustering quality, our algorithm used same privacy budget as that of the existing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Related Work The concept of differential privacy has inspired a plethora of studies, particularly in the area of differentially private k-means clustering [6][7][8] in an interactive setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The important mechanisms of DP in the literature include: the Laplace mechanisms (LapDP) [9], the exponential mechanisms (ExpDP) [10], and the sample and aggregate framework [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' To achieve differential privacy, many im- plementations included infusing Laplace noise into each iteration of Lloyd’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The proportion of noise added was based on a fixed privacy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Some of the strategies for allocating privacy budget included splitting the overall privacy budget uniformly to each iteration [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' However, this requires us to calculate the number of iterations for the convergence, prior to the execution of algorithm, thus increasing the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Fur- ther, researchers overcome this weakness by allocating theoretically guaranteed optimal allocation method [6], but the major assumption taken in this approach was that every cluster has the same size, which does not align with the real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' In another work, Mohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' [8] proposed GUPT, which uses Lloyd’s algorithm for local clustering of each bucket where the items were uniformly sampled to different buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The final result was the mean of locally sampled points in each bucket with added Laplace noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' But, the clustering quality of GUPT was unsatisfying because a large amount of noise was added in the aggregation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Based on the study of past literature on differentially private k-means clustering, Zhigang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' [3] concluded that convergence of an iterative algorithm is important to the clustering quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' To solve this, they introduced the concept of the convergent zone and orientation controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' With the help of a convergent zone and orientation con- troller, they further create a sampling zone for selecting a potential centroid for the 𝑖𝑡ℎ iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The approach iteratively adds noise with an exponential mechanism (ExpDP) by using prior and future knowledge of the po- tential centroid at every step of Lloyd’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The ap- proach maintains the same DP requirements as existing literature, with guaranteed convergence and improve- ment in clustering quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' However, their algorithm perturbs the centroids in a random direction from the center of the cluster, degrading the quality of clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Thus, in this work, we further build upon the approach and significantly improve the clustering quality with the same epsilon privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Preliminaries The definitions used in this work are briefly discussed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The following is a formal definition of Differential Privacy: Definition 1 (𝜖-DP [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' A randomised mechanism T is 𝜖- differentially private if for all neighbouring datasets 𝑋 and 𝑋′ and for an arbitrary answer 𝑠 ∈ 𝑅𝑎𝑛𝑔𝑒(𝑇), T satisfies 𝑃𝑟[𝑇(𝑋) = 𝑠] ≤ 𝑒𝑥𝑝(𝜖) · 𝑃𝑟[𝑇(𝑋′) = 𝑠], where 𝜖 is the privacy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Here, 𝑋 and 𝑋′ differ by only one item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Smaller val- ues of 𝜖 imply a better privacy guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' It is because the difference between the two neighboring datasets is reflected by the privacy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' In this work, we use the ExpDP and LapDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' In exponential DP for non-numeric computation, they introduce the concept of scoring func- tion 𝑞(𝑋, 𝑥), which represents the effectiveness of the pair (𝑋, 𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Here 𝑋 is the dataset and 𝑥 is the response to the 𝑞(𝑋, 𝑥) on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The formal definition of Exponential DP mechanism is defined as follow: Definition 2 (Exponential Mechanism [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Given a scoring function of a dataset 𝑋, 𝑞(𝑋, 𝑥), which reflects the quality of query respond x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The exponential mechanism T provides 𝜖-differential privacy, if 𝑇(𝑋) = {𝑃𝑟[𝑥] ∝ 𝑒𝑥𝑝( 𝜖·𝑞(𝑋,𝑥) 2Δ𝑞 )}, where ∆𝑞 is the sensitivity of scoring function q(X,x), 𝜖 is the privacy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Definition 3 (Convergent & Sampling Zones[3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' A region whose points satisfies the condition: { Node S: ‖𝑆 − 𝑆𝑖 (𝑡)‖ < ‖𝑆𝑖 (𝑡−1) − 𝑆𝑖 (𝑡)‖} is the convergent zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 𝑆𝑖 (𝑡) is defined as the mean of 𝐶𝑖 (𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' A sub-region inside convergent zone is defined as a sampling zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Definition 4 (Orientation Controller[3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 𝑋𝑖 (𝑡) is a direction from the center of the convergent zone to a point on its circumference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' This is the direction along which the center of the sampling zone will be sampled, defined as the orientation controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Approach In this section, we explain our proposed approach and the baseline approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Overview - KMeans Guarantee (Baseline) We took "Differentially Private K-Means Clustering with Convergence Guarantee" [3] as our baseline and im- proved the clustering quality by further building on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Figure 1: Overview of KMeans Guarantee Approach The key concept of the algorithm is to use ExpDP to in- troduce bounded noise into centroids at each iteration of Lloyd’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The technique is designed in a way that it ensures the new centroid is different from the cen- troid of Lloyd’s algorithm while maintaining constraint given in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The constraint guarantees that the perturbed centroid will eventually converge with the centroid of Lloyd’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Their algorithm has four main steps to update the centroids at each Lloyd step t [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The overview of their approach can be seen in (Figure : 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Let the differentially private centroid at iteration 𝑡−1 for a cluster 𝑖 be 𝑆𝑖ˆ (𝑡−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Using this centroid, run one iteration of Lloyd’s algorithm to get the current Lloyd’s centroid 𝑆𝑖 (𝑡) for each cluster 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Using 𝑆𝑖 (𝑡) and 𝑆𝑖 (𝑡−1), generate a conver- gent zone for each cluster 𝑖 as described in 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Generate a sampling zone in the convergence zone and an orientation controller 𝑋𝑖 (𝑡) for each cluster i as defined in 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 3 𝑎𝑛𝑑 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Sample a differentially private 𝑆𝑖ˆ (𝑡) with ExpDP in the sampling zone generated in step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The definition for the convergent zone (for convergence guarantee) and sampling zone (for centroid updating) is defined in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Overview - SubCluster Guarantee We build upon the KMeans Guarantee algorithm to achieve better clustering quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Our idea differs from the baseline in terms of creating a sampling zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' For each cluster, we execute Lloyd’s algorithm over its con- vergent zone to generate its sub-clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Further, we assign each sub-cluster with a probability linearly pro- portional to the number of points it contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Finally, we sample the sub-cluster based on the assigned probability and define it as the sampling zone of the convergent zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Drawing analogy from the KMeans Guarantee algorithm, our orientation controller is this sub-clustering and sam- pling technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Intuitively, our algorithm ensures that Algorithm 1: Differentially Private 𝑘−Means SubClustering Algorithm Input: X = {𝑥1, 𝑥2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='., 𝑥𝑁}: Dataset with N data points k: number of clusters 𝜖𝑒𝑥𝑝: ExpDP privacy budget 𝜖𝑙𝑎𝑝: Laplacian privacy budget for the converged centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙𝐾: number of sub-clusters per cluster Output: S: Final clustering centroids 1 Select 𝑘 centroids S(0) = (𝑆(0) 1 , 𝑆(0) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=', 𝑆(0) 𝑘 ) uniformly from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 2 𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝐹𝑜𝑟𝐿𝑙𝑜𝑦𝑑 = number of iterations to run the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 3 for iters i in 𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝐹𝑜𝑟𝐿𝑙𝑜𝑦𝑑 do 4 for each Cluster i at Iteration t do 5 𝐶(𝑡) 𝑖 ← assign each 𝑥𝑗 to its closest centroid 𝑆𝑖 𝑡−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 6 𝑆𝑖 𝑡 ← centroid of 𝐶𝑖 𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 7 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒𝑖 (𝑡) ← List of data points inside the spherical region having 𝑆𝑖 𝑡 and 𝑆𝑖 𝑡−1 as the endpoints of its radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 8 𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑖 (𝑡) ← run Algorithm 2 using 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒𝑖 (𝑡) , 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 9 𝑆𝑖ˆ (𝑡) ← sample from 𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑖 (𝑡) using ExpDP with 𝑞 and 𝜖𝑒𝑥𝑝;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 10 𝑆𝑖 (𝑡) ← 𝑆𝑖ˆ (𝑡) 11 Publish: 𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑖 (𝑡), 𝑞, 𝜖𝑒𝑥𝑝, 𝑆𝑖 (𝑡) 12 S ← add laplace noise with 𝜖𝑙𝑎𝑝 to S(𝑡);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Algorithm 2: SubClusterSamplingAlgorithm Input: ConvergentZone: Convergent Zone internalK: Subclustering K Output: 𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑡 𝑖 1 S(𝑡): Mean of 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒𝑖 (𝑡) 2 ConvergentZoneClusters ← Cluster ConvergentZone using Lloyd’s algorithm and 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙𝐾 3 ConvergentZoneProbability ← Assign probabilities to the ConvergentZoneClusters proportional to the number of points inside each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 4 SamplingZonei (t) ← Sample a cluster from the ConvergentZoneClusters using ConvergentZoneProbability 5 Return: SamplingZonei (t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Cluster i at Iteration t ( 415 sampling zone (）人 α(t+1) convergence zone orientation controllerFigure 2: Overview of SubCluster Guarantee Approach the sampling zone lies towards the region containing a higher number of data points in an expected case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' With this, we guarantee that our differentially private centroid moves in the direction where the number of data points is higher, incorporating the intuition of Lloyd’s algorithm without compromising on the 𝜖-differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The probability of a differentially private centroid at 𝑖 − 1𝑡ℎ iteration to move in the direction of a more populated re- gion at the 𝑖𝑡ℎ step of Lloyd’s algorithm is also high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Thus, we introduce the concept of sub-clustering in the conver- gent zone and consequently sample one sub-cluster as our sampling zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' We sample the centroid from the sampling zone using the ExpDP mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Finally, we inject Laplace noise in the centroids of the clustering when our algorithm converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' It is because the differentially private cen- troids obtained are a subset of one of the local minima at which Lloyd’s algorithm converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The overview of the proposed approach can be seen in (Figure : 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' We show that a randomized iterative algorithm satisfies an invariant (given in the claim of Lemma 1) and always converges (Proof: refer Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Finally, we show that the SubCluster algorithm is a randomized iterative algorithm that satisfies the invariant(given in Lemma 1) (Proof: Refer Lemma 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' We have four main steps to update the centroids at each Lloyd step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Let the differentially private centroid at iteration 𝑡−1 for a cluster 𝑖 be 𝑆𝑖ˆ (𝑡−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Using this centroid, run one iteration of Lloyd’s algorithm to get the current Lloyd’s centroid 𝑆𝑖 (𝑡) for each cluster 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Using 𝑆𝑖 (𝑡) and 𝑆𝑖 (𝑡−1), generate a conver- gent zone for each cluster 𝑖 as described in 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' SubCluster the convergence zone and sample one of the sub-cluster as our sampling zone based on the probability assigned to each sub-cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The probability assignment is directly proportional to the number of points in each sub-cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Sample a differentially private 𝑆𝑖ˆ (𝑡) with EXpDP in the sampling zone generated in step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Our approach surpasses the baseline approach in terms of clustering quality while maintaining the same DP re- quirements as that of the KMeans Guarantee approach, which is evident from the results obtained (Figure : 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The better clustering quality is a result of our sub- clustering strategy to perturb centroid with a higher prob- ability than the baseline approach towards the direction of the actual centroid generated by Lloyd’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The pseudo-code of our approach is shown in the Algo- rithm 1 and Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Lemma 1: [3] A randomised iterative algorithm 𝜏 is convergent if, in 𝐶(𝑡) 𝑖 (Cluster i at iteration t), 𝑆𝑖ˆ (𝑡)(sampled centroid using 𝜏), 𝑆𝑖 (𝑡−1)(centroid before recentering) and 𝑆𝑖 (𝑡)(centroid of 𝐶(𝑡) 𝑖 ) satisfies the in- variant, ||𝑆𝑖ˆ (𝑡)−𝑆𝑖 (𝑡)|| < ||𝑆𝑖 (𝑡)−𝑆𝑖 (𝑡−1)|| in Euclidean distance, ∀𝑡, 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' We reproduce this lemma from our baseline approach [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Lemma1 and Lemma 2 together provides the com- pleteness and proof for the convergence of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' If the distance between the sampled centroid 𝑆ˆ(𝑡) 𝑖 from the 𝐶(𝑡) 𝑖 and the new centroid 𝑆(𝑡) 𝑖 is less than the dis- tance between the new 𝑆(𝑡) 𝑖 and the old centroid 𝑆(𝑡−1) 𝑖 , then the random iterative algorithm will always converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Intuitively, the loss of 𝐶(𝑡) 𝑖 is minimum if the mean of 𝐶(𝑡) 𝑖 is taken as centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' But, if we slightly shift from the mean of 𝐶(𝑡) 𝑖 , then the loss will increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' However, if we can ensure that any sampled point from 𝐶(𝑡) 𝑖 fulfills the condition: ||𝑆𝑖ˆ (𝑡) − 𝑆𝑖 (𝑡)|| < ||𝑆𝑖 (𝑡) − 𝑆𝑖 (𝑡−1)||, it will lead to a lesser loss than 𝐽𝑆(𝑡−1) 𝑖 , thus, resulting into convergence of the randomised iterative algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' For the mathematical proof, refer [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Lemma 2: Differentially Private 𝑘−Means SubClus- tering approach (SubClustering) is a randomised iterative algorithm that satisfies the invariant ||𝑆𝑖ˆ (𝑡) − 𝑆𝑖 (𝑡)|| < ||𝑆𝑖 (𝑡) − 𝑆𝑖 (𝑡−1)||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Proof: SubClustering is an iterative algorithm that samples a set of centroids for each iteration with Ex- pDP mechanism, thus, making it a randomised itera- tive algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' It subclusters the points lying inside 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡) 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' After subclustering, it samples one subcluster (sampling zone) with the assigned proba- bilities (linearly proportional to the number of data points in subcluster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Finally, it samples a datapoint from the sampled subcluster with ExpDP and call it as the cen- Cluster i at iteration t (C(t) subclusters S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='(t-1) > S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='(t-1) S,(t+7) Srt+s convergent zone ≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' (t) sampling zoneFigure 3: Above figures plots the graph between costGap and epsilon budget for two approaches, the baseline as KmeansGuar- antee and our approach SubClusterGuarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The algorithm was tested on four dataset, Digits (top-left), Wine (top-right), Breast Cancer (bottom-left), and Iris (bottom-right) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' troid of 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡) 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Thus, our sampling zone always lies inside 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡) 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Therefore, the sampled point lies inside 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡) 𝑖 and it sat- isfies the invariant ||𝑆𝑖ˆ (𝑡) − 𝑆𝑖 (𝑡)|| < ||𝑆𝑖 (𝑡) − 𝑆𝑖 (𝑡−1)||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Experimental Setup 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Dataset Used We used following four datasets to test our work Sub- Cluster Guarantee upon the baseline: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Iris [13] dataset comprises total of 150 datapoints with four features and three classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Wine[13] dataset comprises total of 178 data- points with 13 features and three classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Breast Cancer[13] dataset comprises total of 569 datapoints with 30 features and two classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Digits[13] dataset comprises of 1797 datapoints with 64 dimensions and 10 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Metric for Clustering Quality To evaluate the clustering quality, we used the following equation to calculate the normalised difference between the differentially private algorithms (here, SubCluster Guarantee approach) (𝐶𝑜𝑠𝑡𝐷𝑃 ) and Lloyd’s algorithm (𝐶𝑜𝑠𝑡𝐿𝑙𝑜𝑦𝑑): 𝐶𝑜𝑠𝑡𝐺𝑎𝑝 = |𝐶𝑜𝑠𝑡𝐷𝑃 − 𝐶𝑜𝑠𝑡𝐿𝑙𝑜𝑦𝑑| 𝐶𝑜𝑠𝑡𝐿𝑙𝑜𝑦𝑑 (1) The smaller CostGap [3] represents the better quality of clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' In the experiments, we compare the clustering quality of SubCluster Guarantee with KMeans Guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Results and Discussion We tested our algorithm on four datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' All the datasets have different dimensions ranging from 4 to 64 dimen- sions and training sets ranging from 150 to 1800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' As defined in metric smaller gap represents the better clus- tering quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' From the (Figure : 3) we can observe that, cost gap for all the dataset is smaller or equal to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Thus, it is evident that our algorithm has better clustering quality than the existing work for all the datasets experimented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' We varied internalK (parameter for number of sub-clusters) from 2 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Each experiment was conducted 30 times in the case of the Iris, Wine, and Breast cancer dataset and 10 times for digits dataset due to computational constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Fi- nally, for each dataset, we took the average of all the experiments as our final result for plotting the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' KmeansGuaranteevsSubClusterGuaranteeDataset:Digits KmeansGuaranteevsSubClusterGuaranteeDataset:Wine 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='08 KmeansGuarantee KmeansGuarantee 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='175 SubClusterGuarantee 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='07 SubClusterGuarantee 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='05 costGap 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='075 E00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='30 epsilon epsilon KmeansGuaranteevsSubClusterGuaranteeDataset:BreastCancer KmeansGuaranteevsSubClusterGuaranteeDataset:Iris 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='035 14 KmeansGuarantee 12 SubClusterGuarantee 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='030 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='025 KmeansGuarantee 8 SubClusterGuarantee 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='015 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='010 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='30 epsilon epsilonFigure 4: Above figures plots the graph between costGap and epsilon budget for different internalK in SubClusterGuarantee Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The algorithm was tested for internalK=2,3,4,5 for all the four datasets, Digits (top-left), Wine (top-right), Breast Cancer (bottom-left), and Iris (bottom-right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Please note: K and internalK are the same parameter Comparing the SubCluster Guarantee (proposed ap- proach) and K-means Guarantee approach (baseline) by taking an average of all the cost gaps for varied epsilon, and finally taking the ratio between K-means and Sub- Cluster approach: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' In case of Iris dataset, the cost gap is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='1 times smaller than baseline algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' In case of Wine dataset, the cost gap is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='13 times smaller than baseline algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' In case of Breast_Cancer dataset, the cost gap is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='83 times smaller than baseline algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' In case of Digits dataset, the cost gap is almost same as that of baseline algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Detailed Analysis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Iris: Iris dataset has four dimensions and a very small training set of 150 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Our al- gorithm achieves better clustering quality than the baseline algorithm for smaller epsilon values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Since the number of data points is less in Iris, the impact of sub-clustering reduces, resulting in its performance similar to that of the baseline ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' From (Figure : 4), we can observe that changing the value of intenalK has a small impact on the costGap due to a small number of points in each sub-cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' This is because there is a pos- sibility that a sub-cluster has no data point when internalK is increased causing zero probability sub-cluster regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Wine: The wine dataset has 13 dimensions and 178 data points in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Our algorithm performs significantly better than the baseline, as observed in (Figure : 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' It is because the baseline algorithm is constrained to choose a theta in any abrupt direction ranging from [−𝜋/2, 𝜋/2] as shown in (Figure : 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' In contrast, our algorithm shifts the centroids in the direction where the fu- ture centroid of Lloyd’s algorithm is more likely to move (in the expected case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' From (Figure : 4), it is evident that internalK=4 for the wine dataset performs better than the rest of the internalK val- ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Here, the number of dimensions is more than Iris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Therefore, the spatial arrangement will be in an n-sphere which allows better sub-clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Breast_Cancer: Breast_Cancer dataset has 569 data points in its training set and 30 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Our algorithm performs exceptionally better than the baseline, with internalK equal to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' From (Figure : 3), we can observe that there is no monotonous trend for the costGap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Trends are visible in other datasets due to the larger num- ber of classification classes, whereas this dataset has only two classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Thus, adding Laplace noise does not have a relation to the clustering quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Increasing the internalK improves the clustering VaryinginternalKforSubClusteringDataset:Digits VaryinginternalKforSubClusteringDataset:Wine K=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='0035 K=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='20 K=3 K=3 K=4 K=4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='0030 K=5 K=5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='30 epsilon epsilon VaryinginternalkforSubClusteringDataset:BreastCancer VaryinginternalKforSubClusteringDataset:Iris K=2 10 K=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='025 K=3 K=3 K=4 K=4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='020 8 K=5 h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='30 epsilon epsilonquality, with internalK being 4 having the least loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' It is because this dataset has a high number of dimensions and a larger number of training points than other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Digits: It has 64 dimensions and 1797 data points in the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Although it has a large number of dimensions, our algorithm has a very small improvement over the baseline algorithm as seen in (Figure : 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Because of the higher time complexity of our algorithm, it is hard to tune the internalK parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' As the number of sam- ples in a dataset increases, the internalK should increase because a single cluster can contain a large number of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' But, due to limited computational resources, we were not able to ex- periment with it further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' We took internalK to be 5 for our experiments as it performed best in the range [2, 5] as in the (Figure : 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' One of the intriguing findings in the dataset’s results is that the curves based on the internalK have a clearly evident trend, which is a result of the large num- ber of training data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Our proposed algorithm significantly improves over the baseline in terms of clustering quality, especially for the wine and breast cancer dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' In addition our algorithm maintains the same DP requirements as that of existing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Conclusion This work presents a novel method for improving the clustering quality of differentially private k-means al- gorithms while ensuring convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The novelty of our approach is the sub-clustering of the cluster to select the differentially private centroid, which has a higher probability of moving in the direction of the next cen- troid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' We proved that our work surpasses the current state-of-the-art algorithms in terms of clustering quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Especially for the Wine and Breast_Cancer dataset, the clustering quality was significantly improved by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='13 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='83 times than the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' In addition, we maintain the same DP requirements as that of baseline and other existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Future Work In this work, we proved our claim using empirical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' We further plan to validate the results by providing mathematical bounds for the con- vergence degree and rate of the SubClustering Lloyd’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' In terms of clustering qual- ity, the proposed algorithm in this work is com- pared with k-means guarantee clustering only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' to prove the effectiveness of our work, we plan to experiment with other algorithms in the lit- erature including, PrivGene [14], GUPT [8] and DWork [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' The DP requirements in this work are the same as that of past literature, but in the future, we plan to explore ways to improve the current DP guarantees while maintaining the same clustering quality as in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' We used Exponential and Laplace mechanisms of DP in the proposed approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' we further plan to explore the third mechanisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=', sample and aggregate framework, by integrating it with the current algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' In our algorithm, the number of data points inside a cluster is variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Thus we plan to choose an internalK, custom to the size of the cluster to improve the clustering quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'} +page_content=' Acknowledgement We would like to 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100644 index 0000000000000000000000000000000000000000..efbfcfa3b44706bb70361bd38fc3285356e71b85 --- /dev/null +++ b/I9E3T4oBgHgl3EQfXAr0/content/tmp_files/2301.04476v1.pdf.txt @@ -0,0 +1,4159 @@ +1 + +Mid-Infrared Bi-directional Reflectance Spectroscopy of Impact Melt Glasses and Tektites + + +Corresponding Author: Andreas Morlok, Institut für Planetologie, Wilhelm-Klemm-Str. 10, +48149 Münster, Germany. Email: morlokan@uni-muenster.de, Tel. +49-251-83-39069 +Aleksandra Stojic, Institut für Planetologie, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany. +Email: a.stojic@uni-muenster.de +Iris Weber, Institut für Planetologie, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany. Email: +sonderm@uni-muenster.de +Harald Hiesinger, Institut für Planetologie, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany. +Email: hiesinger@uni-muenster.de +Michael Zanetti, University of Western Ontario, 1151 Richmond St, London, Ontario, Canada N6A +3K7. Email: mzanett3@uwo.ca +Joern Helbert, Institute for Planetary Research, DLR, Rutherfordstrasse 2, 12489 Berlin, +Germany, Email: joern.helbert@dlr.de +Keywords: Spectroscopy; Impact processes; Instrumentation; Infrared Observations; Mercury + +© 2016 This manuscript version is made available under the CC-BY-NC-ND 4.0 + + + + + + + + + + + + + + +2 + +Abstract +We have analyzed 14 impact melt glass samples, covering the compositional range from highly +felsic to mafic/basaltic, as part of our effort to provide mid-infrared spectra (7-14 µm) for +MERTIS (Mercury Radiometer and Thermal Infrared Spectrometer), an instrument onboard of +the ESA/JAXA BepiColombo mission. +Since Mercury was exposed to many impacts in its history, and impact glasses are also common +on other bodies, powders of tektites (Irghizite, Libyan Desert Glass, Moldavite, Muong Nong, +Thailandite) and impact glasses (from the Dellen, El’gygytgyn, Lonar, Mien, Mistastin, and +Popigai impact structures) were analyzed in four size fractions of (0-25, 25-63, 93-125 and 125- +250 µm) from 2.5-19 µm in bi-directional reflectance. The characteristic Christiansen Feature +(CF) is identified between 7.3 µm (Libyan Desert Glass) and 8.2 µm (Dellen). Most samples show +mid-infrared spectra typical of highly amorphous material, dominated by a strong Reststrahlen +Band (RB) between 8.9 µm (Libyan Desert Glass) and 10.3 µm (Dellen). Even substantial +amounts of mineral fragments hardly affect this general band shape. +Comparisons of the SiO2 content representing the felsic/mafic composition of the samples with +the CF shows felsic/intermediate glass and tektites forming a big group, and comparatively +mafic samples a second one. An additional sign of a highly amorphous state is the lack of +features at wavelengths longer than ~15 µm. The tektites and two impact glasses, Irghizite and +El’gygytgyn respectively, have much weaker water features than most of the other impact +glasses. +For the application in remote sensing, spectral features have to be correlated with compositional +characteristics of the materials. The dominating RB in the 7-14 µm range correlates well with +the SiO2 content, the Christiansen Feature shows similar dependencies. To distinguish between +glass and crystalline phases of the same chemical composition, a comparison between CF the +SCFM index (SiO2/(SiO2+CaO+FeO+MgO)) (Walter and Salisbury, 1989) is useful, if chemical +compositional data are also available. + +3 + +1. Introduction +This study focuses on mid-infrared reflectance spectra of melt glasses from impact events for the +application in planetary remote sensing. The obtained spectra will be part of a database for the +ESA/JAXA BepiColombo mission to enter the Hermean orbit in 2024 (Maturilli et al., 2008, +Benkhoff et al., 2010), and general remote sensing applications. The ESA/JAXA BepiColombo +mission includes a mid-infrared spectrometer (MERTIS-Mercury Radiometer and Thermal +Infrared Spectrometer) that allows for mapping the surface of Mercury in the 7-14 µm range, +with a spatial resolution of ~500 m (Helbert et al., 2009; Benkhoff et al., 2010; Hiesinger et al., +2010). Laboratory FTIR (Fourier Transformed Infrared) spectra of minerals and rocks, but also +of synthetic analogs, have to be collected to be compared to those from MERTIS for the +interpretation of the remote sensing data from the surface of Mercury (For a discussion of the +comparison of bi-directional reflectance and emission data see below, 2.6 Bi-directional +reflectance FTIR-Analyses). +The surfaces of terrestrial planets and their moons are shaped and modified by impact +events throughout their lifetimes (Hörz and Cintala, 1997). Thus, the investigation of how these +processes affect the spectral properties of the rocks, which are modified or produced during the +impact event is important for the interpretation of infrared data from planetary bodies. In +particular, higher shock ranges result in amorphous phases produced in solid state +transformation (such as maskelynite), or melt glass from complete melting of the target material +(e.g. Stöffler, 1966; Ostertag, 1983; Koeberl, 1986; Dressler and Reimold, 2001; Osinski et al., +2008; Wünnemann et al., 2008; Osinski and Pierrazo, 2012; Jaret et al., 2015a; Pickersgill et al., +2015). +The aim of this study is to provide mid-infrared spectra in the 7-14 µm range for bulk +impact melt rocks, as well as separated glasses for four size fractions (0-25, 25-63, 63-125, 125- +250 µm). Motivation for studying several size fractions is that a high porosity and large grain +size variations are characteristic properties of surface regolith material, which affects the + +4 + +spectral properties of the materials and thus have to be taken into account in the interpretation +of remote sensing data. +Variation in grain size cause the intensity of the characteristic Reststrahlen Bands (RB), +fundamental mode absorption features in the 7-14 µm region, to loose spectral contrast with +decreasing grain size, so the intensity of the bands gets weaker (e.g., Salisbury and Eastes, 1985; +Salisbury and Wald, 1992; Mustard and Hayes, 1997; Ruff and Christensen, 2002). As a +consequence, the identification of phases based on band positions my get difficult. Earlier +studies in the visible and near-infrared indicate a strong influence of grains smaller 30 µm in +sizes for the surface regolith on Mercury (Sprague et al., 2007), so the RB can expected to be +weak in the remote sensing data of this planet. Furthermore, a characteristic additional spectral +feature, the Transparency Feature (TF) appears around 11 – 13 µm in the smallest grain size +fractions as a result of increased volume scattering (e.g. Salisbury, 1993). Such features may +have been observed in earlier, ground based infrared observation of Mercury. Thus laboratory +spectra showing this diagnostic band are also of high interest (e.g. Cooper et al., 2001; Sprague et +al., 2007). + So a series of analyses of distinct size fractions is needed to distinguish between genuine +effects and effects induced by mixed grain size fractions in the natural regolith. However, since +regolith is also an intimate mixture of many phases, spectral unmixing modelling will also have +to take many other grain-size fractions of other potential mineral phases into account, which will +complicate the discussion of the results. +The present work is a follow-up on an earlier study, where we analyzed Suevite impact +rocks from the Nördlinger Ries. The former study shows a high degree of amorphization among +samples of the Suevite impact rock that contains material in all stages of shock metamorphism +(Morlok et al., 2016). Here we focus mostly on impact glasses formed from quenched impacts +melts as representatives of the highest shock stages. In shock metamorphism, mineral phases +start to change above 2 GPa. Planar deformation features appear between 8 and 25 GPa. In + +5 + +addition, microscopic changes in the crystal structure of quartz and feldspar arise. At pressures +between 25 -40 GPa, minerals turn into diaplectic glasses (e.g. maskelynite) in solid-state +transformation, forming of and amorphous material. Onset of melting of phases starts with +feldspar at pressures from ~35 to ~45 GPa, over 60 GPa rocks completely melt. Vaporization +follows at pressures above 100 GPa (e.g., Stöffler, 1966, 1971, 1984; Chao, 1967; von Engelhardt +and Stöffler, 1968; Stöffler and Langenhorst, 1994; French, 1998; Johnson, 2012). However, +these pressures are only approximate to provide a general picture. Shock pressures and +therefore their effects on minerals are often heterogeneously distributed in the rocks (e.g. Hanss +et al., 1978; Johnson et al., 2002). Furthermore, they are based on crystalline, non-porous rocks. +Different abundances of mineral species like quartz and feldspar could affect the exact pressure +range since they show differences in shock transformation (Ostertag, 1983; Stöffler and +Langenhorst, 1984). Petrology (Hörz and Cintala, 1997) but also structural features like grains +size or porosity might play a role (French, 1998; Wünnemeann et al., 2008). Also, variations in +mineral chemical composition are also important. For example, Ca-rich plagioclase transforms +into maskelynite at lower pressures than K- and Na-rich feldspars (e.g. Ostertag, 1983, Johnson +and Hörz 2003, Johnson, 2012). +In order to obtain a more comprehensive understanding of the spectral properties of +highly shocked rocks, we present new mid-infrared spectra of impact glasses originating from a +series of terrestrial impact craters. These glasses are found in the impactites in crater structures, +but are also derived from distant ejecta, the so-called tektites (Dressler and Reimold, 2001). +We selected impact glasses and tektites from a series of terrestrial craters (for all details +see Table 1) in the age range from 0.6 Ma (Lonar crater, Jourdan et al., 2011) to 119 Ma (Mien +impact structure, Bottomley et al., 1978). The size of the respective impact craters or structures +(if known) ranges from 1.8 km for Lonar crater (Wright et al., 2011) to about 100 km for the 36 +Ma old Popigai impact structure (Bottomley et al., 1997; Kettrup et al., 2003). The Canadian +Mistastin Lake impact structure is represented in our study by two melt glass samples (Grieve, +1975; Marion et al., 2011; Pickersgill et al., 2015). + +6 + +The composition of the basement rocks, which are the assumed source material of melts, is +varied, but in most craters dominated by granodioritic and gneissic basement rocks. The +El’gygytgyn impact structure, is an exception with the basement being dominated by rhyolitic +ignimbrites (Layer 2000; Wittmann et al., 2013; Raschke et al., 2014). The Lonar crater formed +in basaltic basement material (Wright et al., 2011). Irghizite from the Zhamanshin impact +structure (Deino et al., 1990) formed in a very diverse target material, consisting of schist, +phyllite, clays and sandstones (Deino et al., 1990; Magna et al., 2011; Ackerman et al., 2015). +A special type of impact glass is called tektite, which is usually very pure, homogenous SiO2-rich +glass with low abundances of inclusions and typically particles display an aerodynamic shape +(e.g., Koeberl, 1986; French, 1998; Dressler and Reimold, 2001). In our study they are +represented by samples from two strewnfields. The Central European strewnfield is represented +by Moldavites. Thailandites and Muong Nong tektites are from the Australasian strewnfield. +Further strewnfields not covered in this study are the Ivory Coast and North American ones +(French, 1998). +In the case of the Australasian tektites (0.77 Ma, Izett and Obradovich, 1992) the large required +source crater (up to 116 km diameter) is unknown (Lee and Wei, 2000; Son and Koeberl, 2005). +Boron isotope studies point towards marine or river sediments as source material (Chaussidon +and Koeberl, 1995). The probable source materials of Moldavites were likely surface sediments +of the Ries impact site, mainly composed of sands and clays (e.g., von Engelhardt et al., 2005). +Impact glass from the Nördlinger Ries crater is represented by two samples. Otting Glasbombe is +glass separated from Suevite originating in the Otting site in the Ries, while the Polsingen sample +is from the Red Suevite, basically a coherent melt rock (Stöffler et al., 2013; Morlok et al, 2016). +Similarly, the source crater for the Libyan Desert glasses is also unknown, but their origin from +an impact into a sand/sandstone type material is strongly indicated by the occurrence of +christobalite, meteoritic components (e.g., chondritic ratios of Co, Cr, Fe, Ni, and Ir) and high- + +7 + +temperature phases like baddeleyite (ZrO2) (e.g., Barnes and Underwood, 1976; Storzer and +Koeberl, 1991; Rocchia et al., 1995; Greshake et al., 2010; Fröhlich et al., 2013). +The mostly granitic and felsic petrology of the crystalline basement of most of the impact +structures is quite different compared to that of Mercury. MESSENGER data indicates that the +best analogs for the surface of Mercury are basalts or ultramafic komatiites (Nittler et al., 2011; +Stockstill-Cahill et al., 2012; Charlier et al., 2013; Maturilli et al., 2014). Unfortunately, naturally +shocked basalts are generally rare (Wright et al., 2011). +Still, a spectral study of shocked felsic material is of interest for our purposes. It gives +insight into the spectral behavior of highly shocked impactites and their components. The data +are also of use for remote sensing of terrestrial impact sites, where impactites with a felsic +mineralogy are common (Wright et al., 2011). Studies of other planetary surfaces than Mercury, +can also benefit from results obtained in this study. For example, granitic material occurs as +fragments and clasts in lunar samples (e.g., Warren et al., 1983; Jolliff et al., 1999; Shervais and +McGee, 1999; Seddio et al., 2015). Glotch et al. (2010) observed evolved lithologies and highly +silicic material (potential lunar granite) in the mid-infrared using the Diviner Lunar Radio +experiment in four lunar regions including the Aristarchus crater, where Mustard et al. (2012) +found high-silica impact melt based on Moon Mineral Mapper data. +On the dominantly basaltic Mars granitoid or felsic materials may occur (e.g., Christensen +et al., 2005; Ehlmann and Edwards, 2014; Sautter et al., 2014, 2015). Furthermore, there are +indications for felsic, granitoid material on Venus (Müller et al., 2008; Gilmore, 2015). +Earlier reflectance and emission studies of impact melt glass and amorphous phases +formed in impacts in the mid-infrared were performed, for example, by Thomson and Schultz +(2002), Gucsik et al., (2004), Faulques et al., (2005) and Palomba et al., (2006). Spectra of these +samples are dominated by a broad feature in the 8.9 to 10.5 µm range, with only few other +features in the mid-infrared. Wright et al., (2011) and Basavaiah and Chavan (2013) analyzed +bulk material, from the basaltic Lonar Crater, India. Jaret et al., (2013, 2015a) analyzed + +8 + +individual minerals using infrared microscopy. All results show a spectrum with a dominating +feature in the ~9.4 - 10.2 µm wavelength range. +Studies on experimentally shocked natural samples (anorthosite, pyroxenite, basalt and +feldspar) were made by Johnson et al., (2002, 2003, 2007, and 2012) and Jaret et al., (2015b). In +these studies, a degradation of features (loss of intensity, band shifts with increasing shock +pressure) were observed due to decreasing structural order. A transmission study of +experimentally shocked feldspars by Ostertag (1983) show similar results. +Similar results with strong features between 9 and 10.5 µm were found by Pollack et al., +(1973); Crisp et al., (1990); Nash and Salisbury (1991) and Wyatt et al., (2001), who studied +terrestrial glasses in basalts or obsidian. +Several mid-infrared reflectance and emission studies used synthetic glasses as +analogues for impact melt glass. Byrnes et al., (2007), and Lee et al., (2010) analyzed synthetic +quartzofeldspathic glasses, finding very good correlations between the band position of +characteristic dominant features and SiO2 contents or Si/O ratios. +Dufresne et al., (2009); Minitti et al., (2002) and Minitti and Hamilton (2010) obtained +comparable results for synthetic glass with basaltic to intermediate composition. Basilevsky et +al., (2000); Moroz et al., (2010) and Morris et al., (2000) measured glass from laser pulse +experiments with Martian soil analogue JSC Mars-1. The resulting spectra are dominated by a +strong single feature in the 9.2-10.5 µm wavelength range. +There are also many studies of melt glasses and highly shocked materials in the visible +and near-infrared. Raikhlin et al., (1987), Schulz and Mustard (2004) and Iancu et al. (2011) +studied terrestrial impact melt rocks, Johnson and Hörz (2003); Adams et al., (1979) and +Bruckenthal and Pieters (1984) experimentally shocked feldspars and enstatite. Moroz et al., +(2009) studied synthetic Martian impact melt analogs and Bell et al. (1976) and Stockstill-Cahill +et al., (2014) analyzed synthetic lunar glass. A study by Keppler (1992) focused on synthetic +silicate glasses with albite and diopside composition. + +9 + +In this study, provide mid-infrared data in the 7-14 µm range of further impact glasses +which were so far not covered by mid-infrared studies. Since the surfaces of many planetary +bodies are covered by regolith, we have to expect intimate mixtures of grains and particles of +different sizes. Grain sizes affect the spectral properties of materials, so this study covers several +grain size fractions (0-25, 25-63, 63-125 and 125-250 µm). +A further question is the identification of spectral parameters, which help to identify +glasses in remote sensing spectra, and to distinguish them from crystalline material with similar +chemical composition. + +2. Material and Methods +2.1 Samples +In order to get representative spectra, we used several grams of melt glass for each of our +analyzed samples. Samples (Table 1) were obtained from the collections of the Institut für +Planetologie (Münster), Mistastin samples were provided by M.Zanetti (St.Louis). Samples of +Lonar crater, Libyan Desert Glass and Irghizite were obtained from meteorite dealers. The Otting +and Polsingen samples were made with material used already in Morlok et al., 2016. + +2.2 Sample Preparation +For the grain size fractions the bulk sample material was first ground in steel and agate mortars +into fine powder. The material was cleaned in acetone and dry sieved for at least one hour into +four size fractions: 0-25 µm, 25-63 µm, 63-125 µm and 125-250 µm, using an automatic Retsch +Tap Sieve. To remove clinging fines, the larger two fractions were again cleaned with acetone. + +10 + +For additional analyses using optical and Scanning Electron Microscopy (SEM), we produced +thick sections of representative blocks of the samples. The sections were polished according to +standard procedures for petrological thin sections, guaranteeing a very flat surface. + +2.3 Optical Microscopy +Overview images of the polished thick sections under normal light were obtained with a +KEYENCE Digital Microscope VHX-500F. Light microscopy provides fast information about the +general homogeneity, amorphous character, as well as enables first mineral identification in the +samples (Fig.1). + +2.4 SEM/EDX +For imaging and the chemical characterization of the samples, we used a JEOL 6610-LV Scanning +electron microscope equipped with an implemented silicon drift Oxford EDX (Energy Dispersive +X-Ray Spectroscopy) system. Images were obtained in the Backscattered Electron modus (BSE) +to enhance contrast due to chemical variations (Fig.2). Chemical analyses with EDX were +quantified with an ASTIMEX™ standard set for major elements. The calibration was confirmed +by re-analyzing the standards in each session. Beam current stability was controlled for each +analysis using a Faraday cup. For measurements of the chemical composition, we routinely +analyzed areas of 100 x 100 µm2 using 90 seconds integration times. A broad beam and shorter +integration times are helpful to measure volatile elements correctly. Results are presented in +Table 2. + + + + +11 + +2.6 Bi-directional reflectance FTIR-Analyses +Each size fraction was gently placed in aluminum sample cups (1 cm diameter, and 1 mm deep). +The surface was flattened with a spatula following a similar procedure described by Mustard +and Hayes (1997) to create a uniform surface without preferred grain orientations. For the +analyses in the mid-infrared from 2.5-19 µm, we used a Bruker Vertex 70 infrared system with a +MCT detector at the IRIS (Infrared and Raman for Interplanetary Spectroscopy) laboratory at the +Institut für Planetologie (Münster). +Most measurements were conducted under low pressure (10-3 bar). In some cases we observed +pore collapse during evacuation even in repeated attempts, resulting in surface distortion. The +probable reason for pore collapse is rapidly expanding volatiles when low pressures are +reached. Attempts to avoid this happening using longer evacuation times or pre-heating to get +rid of adsorbed volatiles were also not successful. So we decided to analyze these sample und +ambient pressure (see Table 3). This may have resulted in water and atmosphere-related +features overlapping with overtone features in the spectral range below 7 µm but is unlikely to +affect longer wavelengths. +To ensure a high signal-to-noise ratio, we accumulated 512 scans for each size fraction. For +background calibration a diffuse gold standard (INFRAGOLDTM) was applied. In order to emulate +various observational geometries of an orbiter, we obtained analyses in a variable geometry +stage (Bruker A513) for the MERTIS database. The data presented here were obtained in a +specular geometry of 30° incidence (i) and 30° emergence angle (e). Results of further analytical +geometries are not presented, since the differences are in the intensities of the spectra, while +effects on band positions were not observed in this study. +Band positions for both powder and microscope analyses of the characteristic features were +obtained using Origin Pro 8. The wavelength of a specific feature was determined by the position +of the strongest reflectance, in the case of the CF we used the position of the lowest reflectance. + +12 + +For the comparison with remote sensing data in the thermal infrared, emission and reflectance +data can be compared using Kirchhoff’s law: ε = 1 – R (R=Reflectance, ε = Emission) (Nicodemus, +1965). However, in this study, a bi-directional, variable mirror set-up was used. This affects the +conversion of the reflectance data to emissivity considerably. Kirchhoff’s’ Law works well for the +comparison of directional emissivity and directional hemispherical reflectance (Hapke, 1993; +Salisbury et al., 1994), for a direct comparison of directional emissivity with reflectance by using +Kirchhoff’s law, the reflected light in all directions has to be collected, using a hemispherical +reflectance, where a gold-coated hemisphere (Salisbury et al., 1991a; Thomson and Salisbury, +1993; Christensen et al., 2001; King et al., 2004). This makes direct quantitative comparisons +between the data obtained in bi-directional mode with surface emission data difficult. For +example, anomalous features heave been observed for the CF. (Salisbury et al., 1991b; Salisbury, +1993; Christiansen et al., 2001). However, similarity in band positions and band shapes, as well +as the low amount of sample materials needed for the bi-directional analyses (which is +important for rare or pure phases) makes this method useful at least for qualitative studies, +when these caveats are kept in mind for direct comparisons with remote sensing data. +The environmental conditions on the surface of Mercury also have to be taken into account. +Temperatures on the surface can reach up to 700K, at a very low ambient pressure below a pico- +bar (e.g. Benkhoff et al., 2010). In contrast, analyses for this study were conducted at low to +ambient pressure, and room temperature. Emissivity studies of olivine, pyroxene and feldspar +under conditions similar to the surface of Mercury or the Moon (e.g. Donaldson Hanna et al., +2012; Helbert et al., 2009 and 2013) show significant shifts in band positions and also a decrease +in spectral contrast for the RB and TF. This also affects direct comparisons between bidirectional +reflectance spectra recalculated to emissivity with the remote sensing data. +The spectral range of interest for the MERTIS database is from 7-14 µm. We present powder +spectra from 7-19 µm (Fig. 3a-c), as features of interest can appear at longer wavelengths. The +signal of the detector used in our study is less effective at wavelengths above 19 µm, resulting in +a low signal-to-noise ratio. Spectra are presented in reflectance, from 0-1. For the spectral range + +13 + +from 2.5-7 µm, where the bands for water are located, we present representative spectra in +Fig.4. +For characterization purposes, we used reference spectra from the Arizona State University +Thermal Emission laboratory (Christensen et al., 2000) and the Johns Hopkins ASTER laboratory +(Baldridge et al., 2009). + +3. Results +3.1. Characterizing the Samples using Optical Microscopy and SEM/EDX +Focus of our chemical analyses was to obtain the composition of the pure glass and that of +inclusions separately. Mistastin Melt 2 shows low totals below 98 wt% (Table 2a). Similar low +totals were observed for Ries glasses in the present and earlier studies (Morlok et al., 2016). +These low totals are probably results of high sample porosities and volatile contents. Chemical +analyses usually confirm earlier results (for references see Table 1). For better identification, +low analytical totals of mineral inclusion were normalized to 100wt% (Table 2b). +The Thailandite and Moldavite tektite samples have all a transparent appearance in optical +microscopy (Fig.1). The Thailandite tektite only shows a small inclusion (~0.5 mm) in the +otherwise nearly pure glass. The inclusion consists of rutile and/or SiO2 polymorphs (Fig.1 and +Table 2b). SEM images also show very homogeneous material (Fig.2). Optical images of +Moldavite reveal dendritic crystal aggregates (Fig.1). Optical images of the Muong Nong sample +(Fig.1) show abundant layers of small vesicles, which are typical for this type of tektite (e.g. +Koeberl, 1992; Son and Koeberl, 2005). Backscattered Electron (BSE) SEM images confirm the +abundant small, empty vesicles (Fig.2). The Libyan Desert Glass has empty vesicles in an +otherwise very homogeneous, colorless glass, with only few small inclusions (Fig.1) of a +crystalline SiO2-polymorph, probably cristobalite, which is typical for this material (Table 2b) + +14 + +(Barnes and Underwood, 1976; Fröhlich et al., 2013). SEM images of the glassy part also show +very homogeneous material (Fig.2). +The El’gygytgyn melt glass shows Schlieren-like structures as well as few mineral inclusions in +the optical images and in the same SEM pictures, embedded in a homogenous, transparent glass +(Fig.1, 2). Inclusions of quartz, magnetite, and ilmenite were observed by Gurov and Koeberl +(2004). The Irghizite impact glass has a brown color, and shows abundant (Koeberl and +Fredriksson, 1986) empty vesicles in the optical and SEM images, but no larger inclusions (Fig.1, +2). +Popigai melt glass also shows a brown color in the optical images, with abundant vesicles and +also some mineral inclusions in the glassy matrix (Fig.1). This was also observed by Whitehead +et al., (2002) and Kettrup et al.,(2003). The inclusions were identified as a SiO2-polmorph with +SEM/EDX, easily distinguishable from the glass in the SEM image by the blocky appearance +(Fig.2, Table 2b). Kettrup et al., (2003) identified Lechatelierite e and coesite in Popigai glass. +The Mien impact melt glass is very heterogeneous, with abundant inclusions in the glassy matrix +(Fig.1). The inclusions were characterized as SiO2-polymorphs and pyroxene by SEM/EDX +(Table 2b, Fig.2). This confirms the findings of Maerz (1979), who observed abundant quartz +and pyroxene mineral fragments. Vesicles also show signs of remnant fillings (Fig.1). +Of the Dellen sample, only powdered material (size >250 µm) was available, so detailed +information about the petrographical context is not available. Deutsch et al. (1992) reported a +glassy matrix with crystallites e.g., pyroxene. This is consistent with bulk chemical analyses +obtained by SEM/EDX in this study (Tab2b). +The samples from the Mistastin lake impact structure have considerable variations. Mistastin +melt 1 shows abundant fragments and vesicles, embedded in a brownish, but translucent glass +(Fig.1, 2). Fragments identified by EDX were SiO2-polymorphs (Tab 2b). Mistastin melt 2 has a +dense, black obsidian-like groundmass with abundant crystallites (also compare Grieve et al., +1975) (Fig.1). The SEM images also show crystallites in a fine-grained matrix (Fig.2). Further + +15 + +phases observed were SiO2-polymorphs, probably quartz (Table 2b) (also see Marion et al., +2010) +The Lonar crater sample is highly vesicular, and the glassy material contains abundant mineral +inclusions, ranging from fragments to clusters of small phases (Fig.1, 2). Identified by SEM/EDX +were SiO2-polymorphs, pyroxene, and ilmenite (Table 2b). This confirms findings of Kieffer et al., +(1976), Osae et al. (2005), and Wright et al., (2011). Impact melt glass from the Lonar crater has +average MgO (3.06wt %) and CaO (6.53wt %) (Table 2a) contents slightly outside the range +observed for melt rocks and breccia in Osae et al., 2005 (5.32 wt% MgO and 8.81 wt% CaO). +Given the heterogeneity and weathered nature of the material, such divergence is expected. + +3.2. Bi-Directional Reflectance FTIR +All impact melt glasses and rocks show similar spectral features in the 7-14 µm wavelength +range (Table 3, Fig.3a-c): A strong RB dominates between 8.9 and 10.3 µm (Features below 7 µm +will be discussed separately in 3.4.1 and Fig.4). The Christiansen Feature (CF), a reflectance +minimum, varies from 7.3 µm (Desert Glass) to 8.2 µm (Dellen)(Table 3). In many samples, the +CF is located at slightly higher wavelengths in the finest size fraction (0-25 µm) when compared +to the coarser ones. The TF usually only occurs in the finest size fraction (0-25 µm) (Table 3). +To aid discussion and interpretation, groupings among the spectra and samples, in addition to +groupings based on chemistry and their source material (tektites, impact glass), were identified +by comparing CF to the SiO2 content (Figure 5). The position of the CF is well correlated with the +SiO2 content of the material (Cooper et al., 2002) (Fig.5). In order to avoid artefacts caused by +low analytical totals resulting from high porosity in a few samples, we compare the SiO2 content +from EDX analyses normalized to 100 wt% totals. +Most of the impact glasses form a group consisting of the felsic/intermediate samples. This +group overlaps with the tektites. Glasses from Mistastin and Lonar, while chemically still at the + +16 + +lower end of the intermediate range (>52 wt% SiO2, Le Maitre, 2002), form the (in relative +terms) ‘mafic’ group. +Silicate glasses show an asymmetric stretching of the Si-O bonds in the 8 – 12.5 µm region, +overlapping with an asymmetric (Si, Al)-O stretching mode near 9 µm. The specific bands are +difficult to distinguish in glass, which results in the characteristic broad band for amorphous +silicates. Further Al-O, Si-O-Si and O-Si-O vibrations occur at longer wavelengths over 11 µm +(McMillan et al., 1998; King et al., 2004, Dufresne et al., 2009, Speck et al., 2011). +In more detail, the group of tektites (Table 3, Fig.3a) shows similar spectral characteristics, a CF +between 7.5 and 7.7 µm, a strong RB from 9.1 to 9.2 µm, and a weak TF between 11.5 and 11.62 +µm. The spectra are very smooth, without significant spectral features of crystalline species. The +intensities of the strongest RB is also very similar at ~0.1 Reflectance (Fig.3a). Also, there are no +significant features at longer wavelengths, except for a weak feature between 12.8 and 13 µm. +The bulk of melt glasses and rocks of the felsic or intermediate groups show a greater spectral +variation that also overlaps with those of tektites (Table 3, Fig3b). El’gygytgyn, Irghizite, Popigai, +Mien, Libyan Desert Glass, Otting, and Polsingen, show simple spectra with shapes similar to +tektites (Fig.3a,b). The spectrum of Libyan Desert Glass is flat at the longer wavelengths as are +the spectra of tektites. Most samples show a shoulder between 7.9 – 8.4 µm, especially +pronounced in the Libyan Desert Glass (Fig.3a-c). Such shoulders or bands are characteristic for +silicate glasses with intermediate composition and related to asymmetric Si-O-Si bridge +vibrations (McMillan and Pirou, 1982; Dufresne et al., 2009). +Positions for the characteristic spectral features range from 7.3 to 7.9 µm for the CF, 8.9 to 9.5 +µm for the strong RB, and 10.9 to 11.9 µm for the transparency feature (Table 3). Intensities +show more variation with the strong RB between 0.05 and 0.1 Reflectance (Fig.3b). Exceptions +are the Libyan Desert Glass with the highest intensity in this study, and the Polsingen sample +with the lowest intensity (Fig.3b). Libyan Desert Glass is the endmember with these strong RB +features occurring at shorter wavelengths, the band positions are also comparable to those of + +17 + +synthetic SiO2 glass (e.g. Faulques et al., 2001). The features in the Ries samples are seen at +longer wavelengths (Table 3). +Furthermore, in most samples in this group weak features are observed in the 8.5-8.8 µm range. +Red Suevite from Polsingen shows several smaller bands between 8.2 and 8.8 µm and also +exhibits stronger bands at longer wavelengths of 17-18.7 µm (Fig.3b), which are probably +caused by the crystalline inclusions. These are typical bands of fragments with granite (or +similar) composition from unmelted crystalline basement rocks (Baldridge et al., 2009, Morlok +et al., 2015). In most cases, at longer wavelengths, there are only weak additional features +comparable to the tektites (Fig. 3), probably result of Al-O, Si-O-Si and O-Si-O vibrations +(McMillan et al., 1998; King et al., 2004). +The samples from Dellen are an outlier (Fig.5). The CF is at 8-8.2 µm, and the TF at 12.1 µm +(Table 3b, Fig3b). The spectra shows no clear main RB, instead it features a ‘twin peak’, band +from 10-10.3 µm which are pyroxene and plagioclase features from fragments (e.g. Baldridge et +al., 2009, Hamilton et al, 2000). The strong additional features at longer wavelengths (>17 µm) +in Dellen and Polsingen are indicative of weathering phases (e.g. clay; Morlok et al., 2016) (Table +2). +Samples from Mistastin and the Lonar craters form the (relatively) mafic group (Table 3, Fig.3c). +The CF falls between 7.9 and 8.1 µm, and the TF is between 11.8 and 12.1 µm. The position of the +dominating RB feature ranges from 9.2 to 10 µm, and the intensities of the strongest RB are +below 0.1 Reflectance. +However, a second feature observed in Lonar at 10.6 µm (Table 3; Fig.3b) could be a shoulder or +feature from unshocked or moderately shocked feldspar (Baldridge et al., 2009; Johnson et al., +2012). Also, comparatively strong features at longer wavelengths (>17 µm; Table 3) hint at +weathering phases (Morlok et al., 2016) (Fig.1, 2; Table 2b). In a similar way, the shoulder of the +Mistastin Melt 2 spectrum at 8.6-8.7 µm and the features from 17.1-18.6 µm (Table 3) can be +explained by the abundant feldspar crystals occurring together with the melt glass, and possibly + +18 + +weathering phases (Fig.1,2) (Grieve et al., 1975; Baldridge et al., 2009; Marion et al., 2010; +Morlok et al., 2016). +In general, the strength of the RB feature between 8.9 to 10.3 µm in all spectra shows of this +study the dominance of amorphous material in the samples. + +3.2.1 Water Bands +The detected water bands were normalized to unity in the 2.5-7 µm regions to allow for better +comparison of the relative intensity of the water feature (Table 3, Fig.4). Although the position of +the water band is very similar in all the samples (it falls between 2.7 and 2.9 µm; e.g. Faulques et +al., 2001), the intensities of the feature vary. However, several of the samples have been +analyzed under ambient pressure, so they could be influenced by atmospheric water. Tektites, as +well as Irghizites and El’gygytgyn show very shallow water bands. Most other samples exhibit +broader bands, indicating adsorbed water in addition to OH groups as part of the mineral +structure. Libyan Desert Glass has a much sharper feature, indicating a very ‘dry’ sample. + +4. Discussion +Results for the strong, dominating RB in impact glasses and tektites in this study (8.9 to 10.3 µm, +Table 3) are similar to those obtained in earlier studies, which also exhibit the strong RB band in +the 8.9 to 10.5 µm range (Thomson and Schultz, 2002; Gucsik et al., 2004; Faulques et al., 2001; +Palomba et al., 2006). In addition, our results are also comparable to terrestrial glasses (9.0-10.5 +µm) Pollack et al., (1973); Crisp et al., (1990); Nash and Salisbury (1991) and Wyatt et al., +(2001). Series of synthetic glasses based on terrestrial rock compositions (Byrnes et al., 2007; +Minitti 2002; DuFresne et al., 2009; Lee et al., 2009; Minitti et al., 2010) show a similar range for +this feature from 9.1 to 10.4 µm (for comparisons with the CF see below). + +19 + +The spectrum of Muong Nong in our study is very similar in band shape and position compared +with results obtained by Gucksik et al. (2004): the strongest RB feature is at 9.2 µm, compared to +an average of 9.2 µm in this study (Table 3, Fig.3a). An Indochinite spectrum with a strong RB at +9.1 µm (Faulques et al., 2001) shows similarity to both Muong Nong (9.2 µm) and Thailandite +(9.1 µm). Another feature near 12.9 µm (Faulques et al., 2001) is found nearby in the results of +our study (Table 3). The dominating RB at 9.1 µm for Moldavite and a weaker band at 12.8 µm is +close to those in Faulques et al. (2001) for the same tektite (9.0 µm and 12.8 µm) (Table 3). A +minor band at 8.1 µm (Faulques et al., 2001) is lacking in our results (Fig. 3a). +The range of results for the main RB in tektites are also comparable to those for synthetic +quartzofeldspathic glass with similar composition in Lee et al., (2010), i.e., 9.1-9.2 µm for this +study (Table 3) compared to 9.1-9.3 µm. The weak water bands are in good agreement with low +water contents (0.002-0.03 wt%) reported in tektites by Beran and Koeberl, 1997 and Dressler +and Reimold, 2001) (Fig.4). Other mid-infrared tektite studies are mostly transmission spectra, +which cannot be directly compared to our reflectance data. However, the typical shape +dominated by a RB feature near 9 µm in tektites is similar to that observed in this study (e.g., +Fröhlich et al., 2013, Morlok et al., 2014). +When compared to synthetic glasses with similar composition (Lee et al., 2010), the range in the +position for the main RB for the felsic/intermediate group in our study is comparable (Fig.6). +Minor features in the 8.5 -8.8 µm and 12.5-12.8 µm ranges observed for most member of this +group can be attributed to crystalline quartz inclusions (e.g. Baldridge et al., 2009). Among the +felsic and intermediate impact glasses, end member Libyan Desert Glass has very similar +features like FTIR analyses by Faulques et al., (2001), which show RB at 7.9, 8.9 µm, and 12.8 µm +(Fig.3b). In contrast, the spectra of Gucsik et al., (2004) has only one RB at 9.6 µm, and another +weaker feature at 14.4 µm. These differences are indicative of the compositional heterogeneity +of the Libyan Desert Glasses. + +20 + +In the felsic and intermediate group, Irghizite and El’gygytgyn samples have weak water bands +like previously seen in the tektites (Beran and Koeberl, 1997) (Fig.4). This indicates very high +temperatures during their formation that dehydrated the material, or low abundance of water in +the starting material. The stronger water bands of the remaining samples in this group could +either point toward higher water contents in the target material, or the influence of weathering. +This was also observed for impactites from the Nördlinger Ries crater, where over 2 wt % H2O +was observed in some glasses (Vennemann et al., 2001; Morlok et al., 2016). Similarly, elevated +water (or volatile) contents of > 1 wt% were found in melt rock and glass from Dellen (Deutsch +et al., 1992), Mien (Schmidt et al., 1997) and Popigai (Kettrup et al., 2003). In addition, the +elevated water content of the Libyan Desert Glass compared with tektites was also observed by +Faulques et al., (2001). +Results from Dellen samples are an outlier. Based on its SiO2 content, the RB and CF should be at +shorter wavelengths (Fig.5). However, the positions are not surprising given the number of +crystalline inclusions in the glass. For example, the observed high abundance of pyroxenes +(Table 2) could have moved the band position of the feature to longer wavelengths (Salisbury, +1991a; Deutsch et al., 1992). +In the more ‘mafic’ group (compared to the previously discussed felsic/intermediate group), the +spectrum of the Lonar samples exhibit the strongest RB at 9.2 µm (Table 3), similar to findings of +Wright et al., (2011), which have the RB at 9.4-9.5 µm for samples showing transformation to +maskelynite (class 2) to complete melting (class 5) (Kieffer et all, 1976; Wright et al., 2011) +(Fig.3c). Jaret et al. (2015a) identified a similar CF position at ~8.0 µm (8.0-8.1 µm in this study, +Table 3). On the other hand, in-situ analyses of completely amorphous material from Lonar have +the main feature at 10-10.4 µm (depending on grain orientation), while a feature at 9.3 µm +occurs in spots which retained some crystallinity (Jaret et al., 2013, 2015a). Differences between +the spectra can be explained with the highly heterogeneous character of the materials resulting +in high contents of fragments (Fig.1, 2) (Kieffer, 1976; Osae et al., 2005; Wright et al., 2011). The + +21 + +normalized water feature of the Lonar material is comparatively weak, the intensity is located +between that of most samples and the tektites (Fig.4). +The two Mistastin melts show slightly different spectra (Fig.3c), likely due to the degree of +crystallinity in the different samples. Mistastin Melt 2 has additional features (8.6-8.7 µm, 17.1- +18.6 µm (Table 3) explained by the abundant crystals within the melt glass (Fig.1,2) (Grieve et +al., 1975; Marion et al., 2010). Mistastin Melt 1, on the other hand, has a spectrum typical for +entirely amorphous materials (e.g. Lee et al., 2010). The position of the strong RB (9.6-9.7 µm, +Table 3) is similar to that of synthetic glasses (~9.8 µm, Lee et al., 2010). The water bands are +comparable to those for the other studied melt glasses (Fig. 4). + +4.1. Identification of Glass +Glasses are characterized in the context of this study by a broad main RB in the 8.9 to 10.3 µm +range (Table 3), in contrast to the usually more feature-rich crystalline felsic and intermediate +rocks (Salisbury, 1988). A comparison of this dominant RB with SiO2 content shows similar +characteristics like the comparison of the CF and the SiO2 content (Fig.6). With increasing SiO2 +content, the band position of the RB moves to shorter wavelengths (also compare Cooper et al., +2002; Lee et al., 2010). However, in some cases remote sensing data only provides spectra with +weak RB features e.g. in many ground based telescope observation in the mid-infrared of +Mercury (e.g. Tyler et al., 1988, Emery et al., 1998, Cooper et al., 2001, Sprague et al., 2002, +2007). So, additional spectral features to identify glassy material are of interest. +If the CF as reflectance minimum (or emission maximum) is obtained in such observations, it is +comparatively easy to identify in remote-sensing data. In a direct comparison between CF and +SiO2-content (Fig.5), the results of our study plot on a slope very similar to that observed for a +series of powdered crystalline plutonic rocks in earlier works (Cooper et al., 2002), with only the +exception of the Dellen and the Libyan Desert Glass data (Fig.5). Thus, on the basis of this type of +diagram, it would be difficult to distinguish melt glass from crystalline material. The + +22 + +independence of the CF from degree of crystallinity was already observed e.g. for feldspar by +Nash and Salisbury, 1990. +An alternative is the comparison between the CF and the SCFM (SiO2, CaO, FeO, MgO) index. The +abundance of divalent cations Ca, Fe, and Mg affects the depolymerization of the silicate +tetrahedra. The SCFM index is calculated from the oxides of Si, Ca, Fe, and Mg: +SiO2/(SiO2+CaO+FeO+MgO) (Walter and Salisbury, 1989; Cooper et al., 2002). The results (Fig.7) +show essentially a similar grouping as in Fig.5. However, impact melt glasses differ slightly from +the trend line for powdered crystalline rocks with comparable chemical composition using the +SCFM index (Fig.7), with most results plotting below the trend line. The SCFM is sensitive to the +degree of polymerization of the rocks (Cooper et al., 2002), while glasses have a low degree of +ordering with only short range structures present. This could explain the divergence, and so +SCFM values plotting below the line for crystalline rocks are possibly indicative of melt glasses. +However, in a remote sensing situation the divergence would require additional chemical data +for the observed material in order to differentiate between glass and crystalline material. +Chemical data for surface regions on Mercury was calculated form data obtained by the X-Ray +Spectrometer (XRS) and the Gamma-Ray Spectrometer (GRS) on the MESSENGER probe that +orbited Mercury (Charlier et al., 2013; Stockstill-Cahill et al., 2013; vander Kaaden et al., 2015; +Peplowski et al., 2015). The BepiColombo mission will include MGNS (Mercury Gamma-ray and +Neutron Spectrometer) and MIXS (Mercury Imaging X-ray Spectrometer) which will provide +chemical data in direct alignment with the mid-infrared data from MERTIS (Benkhoff et al., +2010). +A comparison with micro-FTIR studies of synthetic glass of felsic, intermediate, and basaltic +composition (Fig.6) (Lee et al., 2009) shows a difference between the natural impact melt glass +and the synthetic material. Since the general chemistry of the natural and synthetic glasses is +comparable, effects of different analytical techniques or sample type cannot be ruled out. Cooper +et al. (2002) observed differences between analyses of powdered and solid rocks. On the other + +23 + +hand, Klima and Pieters (2006) observed no differences regarding band positions between these +types of material. +Additional spectral information in the near-infrared could also help to distinguish between +crystalline and glassy material (e.g. Gaffey et al., 1993). In the case of remote sensing of Mercury, +such data is also available from the MESSENGER mission, and will also be provided by +BepiColombo with the Visible Infrared Hyperspectral Imager Channel (VIHI) on the +Spectrometer and Imagers for BepiColombo Integrated Observatory System (SIMBIO-SYS) +instrument suite (Benkhoff et al., 2010). + +4.2. Application to MERTIS and Remote Sensing of Mercury +Mid-infrared data from Mercury is rare due to difficulties for ground-based observations (e.g. +Cooper et al., 2001). Spectral studies by ground- and airplane based telescopes in the mid- +infrared were made by Sprague et al. 1994, 2000, 2002, 2007; Emery et al. 1998; Sprague and +Roush 1998; Cooper et al. 2001; Donaldson-Hanna et al. 2007). Due to observational limitations, +they all cover large surface areas from at least 104 - 106 km2. The dominating mineral phases +identified in these studies are mainly plagioclase with minor pyroxene (e.g. Sprague et al., 2007). +Most spectra from Mercury show only weak features and low spectral contrast and probably +have a low signal to noise ratio. Potential CF are visible from 7.7 – 8.7 µm. The CF observed in +this study overlap with this range at shorter wavelengths, with the glasses from the ‘mafic’ +group, Dellen and some from the intermediate group falling into the range observed for the CF +on Mercury (Tab.3). The TF for the observations of Mercury is between 12 and 12.7 µm, only +partially overlapping with the TF in our study (10.9 – 12.1 µm). Only Dellen, Mistastin and Lonar +samples have CF in this region (Tab.3) (Sprague et al. 1994, 2000, 2002, 2007; Emery et al. 1998; +Sprague and Roush 1998; Cooper et al. 2001; Donaldson-Hanna et al. 2007). RB are difficult to +identify in the surface spectra. Candidates are found mostly at 9.2-9.5 µm (Emery et al., 1998; + +24 + +Sprague et al., 1994; Sprague et al., 1998; Sprague et al., 2002; Donaldson-Hanna 2007) would +overlap with most of the strong RB in the intermediate and mafic group (Tab.3). +Since these areas observed so far are from vast surface regions, the spectral features of distinct +regions with characteristic compositions are integrated into one spectrum. This makes it +difficult to discuss the results in a wider context, such as smaller surface features. Here the high +spatial resolution of MERTIS of ~500 m (Benkhoff et al., 2010) will allow to resolve smaller +structures such as the hollows (e.g. Thomas et al. 2014). +However, regolith is a mixture of many phases, so glass would be only one of several +mineral/glass components and size fractions, which have to be taken into account in spectral +deconvolution modelling. Also, the conversion problems (See 2.6 Bi-directional reflectance FTIR- +Analyses) of bi-directional reflectance for comparison with emissivity data will have to be taken +into account for direct comparisons of laboratory and remote sensing data. + +5. Summary and Conclusions +We have characterized and analyzed mid-infrared spectra of four size fractions (0-25, 25-63, +93-125 and 125-250 µm) from 14 impact melt glass samples, covering the compositional range +from highly felsic (Libyan Desert Glass) to mafic/basaltic (Lonar crater melt). +Most samples show mid-infrared spectra typical of highly amorphous material, dominated by a +strong Reststrahlen Band between 8.9 and 10.3 µm (Table 3). Even substantial amounts of +mineral fragments hardly affect this general band shape. However, crystallization from the melt, +such as in one of our two Mistastin samples (Mist Melt 2), can result in significant differences in +the spectra between compositionally similar samples. + + +25 + +An additional sign of a highly amorphous state is the lack of features at wavelengths longer than +~15 µm (Fig.3a-c). Tektites have much weaker water features than most of the other impact +melt glasses, with the exception of samples from Irghizite and El’gygytgyn. +For the application in remote sensing, spectral features have to be correlated with compositional +characteristics of the materials. A convenient method to correlate compositional and spectral +features is to use the prominent and in spectra with low signal/noise ratio easily recognizable +Christiansen Feature and compare it to compositional parameters like the SCFM index (Cooper +et al., 2002). The comparison shows some differences, which could help distinguish glass from +crystalline material in remote sensing, using chemical compositional data from further +instruments onboard of BepiColombo. +A comparison between the laboratory spectra and mid-infrared ground-observations of Mercury +shows similarities in band positions between various features. 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Tektites Thailandite, +Muong Nong and Moldavite can be recognized by their transparent appearance and very low +content of inclusions. Most other melt glasses show higher abundances of crystalline material +e.g. Popigai, Mien), while Libyan Desert glass has an appearance similar to the tektites. Samples +like Lonar or Mistastin Melt 2 have very high contents in inclusions. +Figure 2. SEM/BSE images of polished sections from the samples analyzed in this study. The +findings of the optical microscopy are mainly confirmed, high contents of vesicles are visible in +the Muong Nong, Irghizite and Lonar samples. +Figure 3. Mid-infrared bi-directional reflectance spectra of impact melt and glass samples. Band +positions in µm. In reflectance (0-1). Blue: 0-25 µm, Pink: 25-63 µm, Red: 63-125 µm, Brown: +125-250 µm (in µm). (a) Tektites, (b) Felsic/intermediate Samples, (c) Mafic/basaltic samples. +Figure 4. Spectral range from 2.5 to 7 µm for each sample (always size fraction 125-250µm, +normalized on unity), which shows the water features at ~3 µm. Spectra are offset along the y- +axis for clarity. +Figure 5. Comparison of SiO2 concentration (in wt%) in the samples with the position of the +Christiansen Feature (in µm). Dotted line: Crystalline felsic and intermediate rocks from Cooper +et al. (2002). Most samples from this study form a group of felsic/intermediate material +(including the tektites), with a small group of near-mafic samples (Mistastin, Lonar) forming the +‘mafic’ group. +Figure 6. Comparison of SiO2 concentration (in wt%) in the samples with the position of the +strongest Reststrahlen Band (in µm). The dotted line is the trend line for the synthetic glass with +intermediate to felsic composition (Lee et al., 2010). Most results for the glasses differ from the +results for synthetic glass (DuFresne et al., 2009; Lee et al., 2010). + +43 + +Figure 7. Comparison of the SCFM-index (SiO2/(SiO2+CaO+FeO+MgO) (Walter and Salisbury, +1989, Cooper et al., 2002) with the position of the Christiansen Feature (in µm). Dotted line: +Crystalline felsic and intermediate rocks from Cooper et al., (2002). The results for the impact +glass mostly plot below the trend line for crystalline material. + + + +1 + + +Source +Sample +Age +(Ma) +Size +crater +(km) +Source rock +Lit. +Australasian +Strewnfield +Muong +Nong + +0.8 +90- +116? +Marine/River +sediments? +[1,2,3,4] + +Australasian +Strewnfield +Thailandite/ + +0.8 +90- +116? +Marine/River +sediments? +[1,2,3,4] +Nördlinger +Ries Crater +Moldavite +14.3 +26 +Sediments, sands, +clays +[5,6,7,8,9] + +Otting +Polsingen +14.3 +26 +Granite, Gneiss, +Amphibolite +[10] +El’gygytgyn +Impact +Structure +El’gygytgyn +3.58 +18 +Rhyodacitic/Rhyolitic +Ignimbrite +Basalt, Andesite +[11,12,13] + +Zhamanshin +Impact +Structure +Irghizite +0.9 +5.5-6.3 +Quartz-sericite schist, +phyllite , clays and +locally sandstones, +Ultrabasic Intrusion +[6,14,15, +16,17] +Popigai +Impact +structure +Popigai +35.7 +100 +Gneiss +[18,19,20] +Mien Impact +Structure +Mien +118.7 +9 +Granite +Gneiss +Amphibole +[21,22] +? +Libyan +Desert Glass +29 +? +Sands/Sandstone +[5,23] + + + + + + + +Dellen +Impact +Structure +Dellen +89 +20 +Granodiorite +Gneiss +[21,24] + + + + + + +Mistastin +Lake Impact +Structure +Mistastin +36 +28 +Granodiorite, +Anorthosite, +Mangerite +[25,26,27,28] + +Lonar Lake +Crater +Lonar +0.57 +1.8 +Basalt +[29,30,31] + + +Table 1. Overview of the samples and their sources used in this study. Age in million years (Ma), +Crater size in km, Basement rock: petrology of probably source material for melts. Lit.: literature +sources for data: [1] French 1998, [2] Chaussidon and Koeberl 1995, [3] Izett and Obradovich +1992, [4] Lee and Wei 2000, [5] Magna et al., 2011, [6] Koeberl and Fredriksson., 1986, [7] +Laurenzi et al., 2003, [8] Buchner et al., 2003, [9] von Engelhardt et al., 2005, [10] Stöffler et al., +2013, [11] Layer et al., 2000, [12] Raschke et al., 2014, [13] Gurov and Koeberl 2004, [14] +Koeberl , 1986,[15] Taylor et al., 1979, [16] Ackerman et al., 2015, [17] Deino et al., 1990, [18] +Bottomley et al., 1997, [19] Withehead et al., 2002, [20] Kettrup et al., 2003, [21] Schmidt et al., +1997 [22] Bottomley et al., 1978, [23] Fröhlich et al., 2013, [24] Deutsch et al.,1992, [25] Marion +et al., 2010, [26] Grieve 1975, [27] Marchand et al., 1977, [28] Pickersgill et al., 2015, [29] Osae +et al., 2005, [30] Wright et al., 2011, [31] Jourdan et al., 2001. + + +2 + +Table 2a. Chemical Composition of the impact melt rocks and glasses. SEM/EDX data, in wt%. Results for Otting and Polsingen are from Morlok et al., +2016. s.d.=Standard Deviation (1Ω). Mist.=Mistastin. Elgy.= El’gygytgyn. + + +Muong +Nong + + +s.d. + +Thai- +landite + + +s.d. + +Molda- +vite + + +s.d. + + +Elgy + + +Irghizite + + +s.d. + + +Popigai + + +s.d. + + +Mien + +Desert +Glass + + +s.d. + + +Otting + + +s.d. + + +Polsingen + + +s.d. + + +Dellen + + +s.d. +Na2O +1.39 +±0.01 +1.24 +±0.01 +0.43 +±0.02 +2.92 +1.03 +±0.02 +2.45 +±0.03 +3.18 +0.03 +±0.02 +2.79 +±0.27 +3.23 +±0.31 +2.83 +±0.57 +MgO +1.96 +±0.12 +1.96 +±0.02 +1.53 +±0.04 +0.82 +2.92 +±0.04 +4.00 +±0.02 +0.38 +0.01 +±0.01 +2.82 +±0.26 +0.33 +±0.14 +0.20 +±0.01 +Al2O3 +14.80 +±0.20 +13.17 +±0.09 +9.82 +±0.08 +15.25 +9.51 +±0.08 +16.90 +±0.10 +14.83 +1.15 +±0.63 +14.99 +±0.42 +17.91 +±0.75 +12.09 +±0.25 +SiO2 +70.65 +±0.22 +72.70 +±1.00 +79.81 +±0.59 +70.79 +73.66 +±0.37 +60.78 +±0.22 +69.82 +99.07 +±1.19 +59.84 +±1.10 +56.37 +±0.84 +74.56 +±0.37 +P2O5 +n.d. + +n.d. + +n.d. + +0.08 +0.06 +±0.04 +0.12 +±0.03 +0.16 +n.d. + +0.39 +±0.04 +0.46 +±0.08 +0.24 +±0.04 +SO3 +0.09 +±0.06 +0.06 +±0.04 +0.08 +±0.07 +0.10 +0.05 +±0.04 +0.35 +±0.08 +0.09 +0.13 +±0.02 +0.06 +±0.02 +0.06 +±0.04 +0.03 +±0.02 +K2O +2.84 +±0.02 +2.44 +±0.06 +3.43 +±0.03 +4.17 +1.91 +±0.01 +2.94 +±0.02 +5.29 +n.d. + +3.98 +±0.25 +5.85 +±0.59 +4.85 +±0.41 +CaO +1.33 +±0.15 +1.89 +±0.05 +1.97 +±0.06 +2.64 +2.38 +±0.01 +2.97 +±0.03 +1.83 +n.d. + +3.65 +±0.25 +3.37 +±0.42 +0.94 +±0.17 +TiO2 +0.86 +±0.03 +0.76 +±0.04 +0.37 +±0.02 +0.35 +0.76 +±0.04 +0.87 +±0.03 +0.40 +0.17 +±0.06 +0.93 +±0.05 +1.03 +±0.05 +0.49 +±0.02 +Cr2O3 +0.04 +±0.02 +0.02 +±0.02 +0.02 +±0.02 +0.05 +0.06 +±0.03 +0.05 +±0.03 +n.d. +0.01 +±0.01 +0.04 +±0.01 +0.02 +±0.01 +n.d. + +MnO +0.11 +±0.01 +0.11 +±0.05 +0.05 +±0.01 +0.08 +0.09 +±0.03 +0.07 +±0.01 +0.03 +0.02 +±0.02 +0.09 +±0.03 +0.01 +±0.01 +0.04 +±0.01 +FeO +5.45 +±0.02 +4.79 +±0.05 +1.73 +±0.03 +3.03 +5.97 +±0.08 +8.00 +±0.02 +1.99 +0.10 +±0.04 +4.81 +±0.29 +2.14 +±0.64 +2.55 +±0.11 +NiO +0.03 +±0.04 +0.01 +±0.03 +0.01 +±0.02 +0.00 +0.18 +±0.02 +0.00 +±0.00 +0.00 +0.02 +±0.02 +0.01 +±0.01 +0.00 +±0.01 +0.04 +±0.03 +SUM +99.56 + +99.14 + +99.24 + +100.28 +98.58 + +99.50 + +98.00 +100.70 + +94.41 + +90.76 + +98.83 + + +Table 2 cont. + +Mist. +Melt +1 + + +s.d. +Mist. +Melt +2 + + +Lonar + + +s.d. +Na2O +4.13 +±0.06 +3.99 +1.28 +±0.13 +MgO +1.22 +±0.02 +1.03 +3.08 +±0.18 +Al2O3 +20.35 +±0.08 +20.56 +14.67 +±1.40 +SiO2 +56.61 +±0.04 +53.54 +57.59 +±1.38 +P2O5 +0.46 +±0.01 +0.41 +0.65 +±0.05 +SO3 +0.04 +±0.01 +0.11 +0.06 +±0.05 +K2O +1.93 +±0.05 +1.51 +1.62 +±0.00 +CaO +7.18 +±0.06 +6.82 +6.53 +±0.42 +TiO2 +1.10 +±0.00 +1.09 +2.02 +±0.23 +Cr2O3 +0.07 +±0.03 +0.05 +0.02 +±0.02 +MnO +0.10 +±0.01 +0.05 +0.16 +±0.06 +FeO +5.99 +±0.14 +5.55 +11.67 +±0.88 +NiO +0.02 +±0.02 +0.02 +n.d. + +SUM +99.16 + +94.73 +99.31 + + +3 + + + +Thai +Rutile +SiO2 +Elgy. + + +Popigai +SiO2 +Mien +Pyx +SiO2 +Libyan +Desert +SiO2 +Dellen + +Mist. +Melt +1 +SiO2 + +Melt. +2 +SiO2 +Lonar +Pyx +SiO2 +Ilmenite +Na2O +0.09 +0.04 +2.88 +0.08 +0.08 +0.05 +0.07 +0.09 +0.05 +0.06 +0.15 +0.13 +0.09 +MgO +0.01 +0.03 +0.93 +0.03 +25.80 +0.07 +n.d +17.56 +0.02 +0.03 +15.91 +0.01 +1.25 +Al2O3 +0.47 +n.d +15.15 +0.26 +1.00 +0.64 +1.69 +1.49 +n.d. +n.d +0.94 +1.69 +0.16 +SiO2 +0.87 +99.19 +70.47 +99.71 +54.21 +98.80 +98.26 +50.93 +98.95 +99.26 +50.43 +95.92 +0.19 +P2O5 +n.d +0.05 +0.07 +0.06 +n.d +0.03 +0.12 +0.03 +0.11 +0.06 +0.07 +0.03 +0.07 +SO3 +n.d +0.04 +n.d +0.11 +0.07 +0.06 +0.09 +0.04 +0.06 +0.11 +0.06 +0.09 +0.02 +K2O +0.03 +n.d +4.07 +n.d +0.01 +n.d +n.d +0.01 +n.d +n.d +n.d +0.71 +n.d +CaO +0.17 +n.d +2.74 +0.03 +1.09 +n.d +n.d +0.93 +n.d +n.d +4.69 +0.38 +0.14 +TiO2 +97.97 +n.d +0.39 +0.04 +0.16 +0.03 +0.21 +0.27 +0.03 +0.01 +0.58 +n.d +51.77 +Cr2O3 +n.d +0.03 +0.05 +0.01 +0.21 +0.00 +0.02 +0.05 +0.02 +n.d +n.d +0.03 +0.23 +MnO +n.d +n.d. +0.10 +n.d +0.59 +0.06 +n.d +0.54 +0.01 +0.06 +0.52 +n.d +0.48 +FeO +1.51 +0.57 +3.11 +0.13 +17.56 +0.25 +0.15 +29.79 +0.05 +0.03 +27.17 +0.05 +45.53 +NiO +n.d +0.06 +0.06 +0.03 +0.19 +n.d +n.d +0.11 +0.01 +n.d +n.d +n.d +0.09 +SUM +101.12 +100.00 +100.00 +100.49 +100.97 +100.00 +100.61 +101.84 +99.31 +99.62 +100.52 +99.04 +100.00 + +Table 2b. Chemical Composition for some inclusions in the impact melts and glass. SEM/EDX data, in wt%. Samples with low totals below 98%wt% were +normalized to 100wt% for better mineral identification. Mist.=Mistastin, Elgy= El’gygytgyn, Libyan Desert=Libyan Desert Glass, Thai=Thailandite, +Pyx=Pyroxene, Qtz=Quartz, Ilm=Ilmenite. Mist.=Mistastin. Elgy.= El’gygytgyn. + + + + +4 + + + + + +CF + + + + + + +TF + + + + + + + + + + + + + + + + +Thailandite + + + + + + + + + + + + + +0-25 +2.8 + +5.42 +7.65 + + + +9.13 + + +11.6 + + +25-63 +2.79 + +5.42 +7.63 + + + +9.12 + + + + +12.83 +63-125 +2.79 + +5.42 +7.56 + + + +9.13 + + + + +12.78 +125-250 +2.79 + +5.45 +7.63 + + + +9.13 + + + + +12.79 + + + + + + + + + + + + + + +Muong Nong + + + + + + + + + + + + + +0-25 +2.8 + +5.4 +7.68 + + + +9.16 + + +11.62 + + +25-63 +2.8 + +5.4 +7.67 + + + +9.15 + + + + +12.97 +63-125 +2.8 + +5.4 +7.67 + + + +9.15 + + + + +12.82 +125-250 +2.8 + +5.5 +7.64 + + + +9.15 + + + + +12.82 + + + + + + + + + + + + + + +Moldavite + + + + + + + + + + + + + +0-25 +2.79 +5.42 + +7.55 + + + +9.07 + + +11.54 + + +25-63 +2.79 +5.41 + +7.54 + + + +9.07 + + + + +12.85 +63-125 +2.79 +5.41 + +7.53 + + + +9.07 + + + + +12.79 +125-250 +2.79 +5.42 + +7.53 + + + +9.07 + + + + +12.79 + + + + + + + + + + + + + + +El’gygytgyn + + + + + + + + + + + + + +0-25 +2.8 +5.5 + +7.71 + + + +9.24 + + +11.62 + + +25-63 +2.8 +5.52 + +7.66 + + + +9.24 + + + + +12.89 +63-125 +2.8 +5.53 + +7.64 + + + +9.24 + + + + +12.88 +125-250 +2.8 +5.57 + +7.64 + + + +9.25 + + + + +12.99 + + + + + + + + + + + + + + +Irghizite + + + + + + + + + + + + + +0-25 +2.79 +5.4 + +7.61 + + + +9.09 + + +11.6 + + +25-63 +2.79 +5.4 + +7.6 + + + +9.08 + + + + +12.77 +63-125 +2.79 +5.4 + +7.61 + + + +9.08 + + + + +12.77 +125-250 +2.79 +5.43 + +7.61 + + + +9.09 + + + + +12.75 + + + + + + + + + + + + + + +Table 3. Band positions of features in powdered impact melts and glasses (in µm). CF=Christiansen Feature, TF=Transparence Feature, bel.: analyzed +under ambient pressure. + + +5 + + + + + +CF + + + + + + +TF + + + + +Popigai + + + + + + + + + + + + + + + +0-25 +2.8 +5.4 +6.15 +7.72 + +8.56 + +9.21 + + +11.45 +13.36 +14.81 + + +25-63 +2.78 +5.4 +6.15 +7.65 + + + +9.24 + + + +12.54 +12.84 + + +63-125 +2.78 +5.4 +6.15 +7.6 + + + +9.21 + + + +12.54 +12.8 + + +125-250 +2.78 + +6.15 +7.62 + + + +9.32 + + + + +12.82 + + + + + + + + + + + + + + + + + + +Mien (bel.) + + + + + + + + + + + + + + + +0-25 +2.75 + +6.1 +7.78 + + + +9.17 + + +11.75 + + +17-17.5 +18-19 +25-63 +2.75 + +6.1 +7.73 + +8.58 + +9.17 + + +11.71 + + +17-17.5 +18-19 +63-125 +2.75 + +6.1 +7.61 + + + +9.15 + + + + + +17-17.5 +18-19 +125-250 +2.75 + +6.1 +7.59 + + + +9.16 + + + + + +17-17.5 +18-19 + + + + + + + + + + + + + + + + +Libyan Desert +Glass + + + + + + + + + + + + + + + +0-25 +2.72 +5.34 +6.17 +7.28 +7.92 + + +8.9 + + +11 + + + + +25-63 +2.72 +5.34 +6.15 +7.29 +8 + + +8.89 + + +10.92 + + + + +63-125 +2.72 +5.34 +6.16 +7.28 +8.03 + + +8.89 + + + + + + + +125-250 +2.72 +5.44 + +7.27 +8.09 + + +8.9 + + + + + + + + + + + + + + + + + + + + + + + +Otting Glasbombe +(bel.) + + + + + + + + + + + + + + + +0-25 +2.75 + +6.1 +7.82 + +8.57 + +9.33 + + +11.76 + + + + +25-63 +2.75 + +6.1 +7.8 + +8.58 + +9.34 + + + + + + + +63-125 +2.75 + +6.1 +7.79 + + + +9.39 + + + + + + + +125-250 +2.75 + +6.1 +7.77 + + + +9.38 + + + + + + + + + + + + + + + + + + + + + + + +Polsingen (bel.) + + + + + + + + + + + + + + + +0-25 +2.76 +2.9 +6.1 +7.88 +8.22 +8.52 +8.72 +9.48 + + +11.81 + + + + +25-63 +2.76 +2.9 +6.1 +7.84 +8.27 +8.56 +8.78 +9.47 + + +11.89 + + +17.0-17.5 +18.7 +63-125 +2.76 +2.9 +6.1 +7.76 +8.32 +8.57 +8.84 +9.48 + + + + + +17.0-17.5 +18.7 +125-250 +2.76 +2.9 +6.1 +7.61 + + +8.84 +9.49 + + + + + +17.0-17.5 +18.7 + + + + + + + + + + + + + + + + +Dellen3/4 (bel.) + + + + + + + + + + + + + + + +0-25 +2.75 + +6.1 +8.19 + +8.75 + +9.66 +10.01 +10.27 +12.09 + + + + +25-63 +2.75 + +6.1 +8.12 + +8.76 + +9.69 +10 + + + + +15.6 + +63-125 +2.75 + +6.1 +8.06 + +8.77 + + +9.95 + + + + +15.7 + +125-250 +2.75 + +6.1 +8.01 + + + + +9.97 +10.31 + + + +15.68 + + + + + + + + + + + + + + + + + +Table 3 cont. + +6 + + + + + +CF + + + + + + +TF + + + + +Mistastin +Melt 1 + + + + + + + + + + + + + + + +0-25 +2.9 + + +7.99 + + + +9.66 + + +11.8 + + + + +25-63 +2.8 + + +7.96 + + + +9.64 + + + + + + + +63-125 +2.8 + + +7.9 + + + +9.66 + + + + + + + +125-250 µm +2,8 + + +7,92 + + + +9.65 + + + +14.31 + + + + + + + + + + + + + + + + + + + +Mistastin +Melt 2 (bel.) + + + + + + + + + + + + + + + +0-25 +2.76 + +6.1 +8.07 + +8.56 + +9.96 + + +12.13 + + +17.07 +18.6 +25-63 +2.76 + +6.1 +8.01 + +8.74 + +9.88 + + +12.05 + + +17.29 +18.43 +63-125 +2.76 + +6.1 +8.02 + + + +9.89 + + + + + +17.28 +18.47 +125-250 +2.76 + +6.1 +7.93 + + + +9.86 +9.68 + + + + +17.25 +18.48 + + + + + + + + + + + + + + + + +Lonar Melt (bel.) + + + + + + + + + + + + + + + +0-25 +2.74 + +6.1 +8.14 + + + +9.18 +10.68 + +12.09 + + + + 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+Reflectance +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +0,00 +0,01 +0,02 +0,03 +0,04 +0,05 +0,06 +0,07 +0,08 +0,09 +0,10 +B +A +Moldavite +Band Position in µm +Reflectance +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +0,00 +0,01 +0,02 +0,03 +0,04 +0,05 +0,06 +0,07 +0,08 +0,09 +0,10 +Muong Nong +Reflectance +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +0-25 µm +125-250 µm +63-125 µm +25-63 µm +1400 +1200 +1000 +800 +600 +Band Position in cm-1 +Figure 3 + +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +0,00 +0,05 +0,10 +0,15 +0,20 +0,25 +B +A +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +0,00 +0,01 +0,02 +0,03 +0,04 +0,05 +0,06 +0,07 +0,08 +0,09 +0,10 +B +A +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +0,00 +0,01 +0,02 +0,03 +0,04 +0,05 +B +A +Reflectance +Mien +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +0,00 +0,01 +0,02 +0,03 +0,04 +0,05 +0,06 +0,07 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Quartzofeldspathic Glass (Lee et al.,2010) +1120 +1080 +1040 +1000 +960 +Strongest Feature (RB) in cm-1 +Figure 6 + +1320 +1280 +1240 +1200 +1160 +1120 +SCFM Index +Christiansen Feature (CF) in µm +0.75 +0.8 +0.85 +0.9 +0.95 +1 +0.7 +7.20 +7.30 +7.40 +7.50 +7.60 +7.70 +7.80 +7.90 +8.00 +8.10 +8.20 +Tektites +Felsic/Intermediate Glass +Dellen +Mafic Glass +Christiansen Feature (CF) in cm-1 +Figure 7 + diff --git a/I9E3T4oBgHgl3EQfXAr0/content/tmp_files/load_file.txt b/I9E3T4oBgHgl3EQfXAr0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2940ebce5db498bd4c054cd8bcd8474a2d8579ce --- /dev/null +++ b/I9E3T4oBgHgl3EQfXAr0/content/tmp_files/load_file.txt @@ -0,0 +1,3465 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf,len=3464 +page_content='1 Mid-Infrared Bi-directional Reflectance Spectroscopy of Impact Melt Glasses and Tektites Corresponding Author: Andreas Morlok, Institut für Planetologie, Wilhelm-Klemm-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 10, 48149 Münster, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Email: morlokan@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='de, Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' +49-251-83-39069 Aleksandra Stojic, Institut für Planetologie, Wilhelm-Klemm-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 10, 48149 Münster, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Email: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='stojic@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='de Iris Weber, Institut für Planetologie, Wilhelm-Klemm-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 10, 48149 Münster, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Email: sonderm@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='de Harald Hiesinger, Institut für Planetologie, Wilhelm-Klemm-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 10, 48149 Münster, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Email: hiesinger@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='de Michael Zanetti, University of Western Ontario, 1151 Richmond St, London, Ontario, Canada N6A 3K7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Email: mzanett3@uwo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='ca Joern Helbert, Institute for Planetary Research, DLR, Rutherfordstrasse 2, 12489 Berlin, Germany, Email: joern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='helbert@dlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='de Keywords: Spectroscopy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Impact processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Instrumentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Infrared Observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Mercury © 2016 This manuscript version is made available under the CC-BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='0 2 Abstract We have analyzed 14 impact melt glass samples, covering the compositional range from highly felsic to mafic/basaltic, as part of our effort to provide mid-infrared spectra (7-14 µm) for MERTIS (Mercury Radiometer and Thermal Infrared Spectrometer), an instrument onboard of the ESA/JAXA BepiColombo mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Since Mercury was exposed to many impacts in its history, and impact glasses are also common on other bodies, powders of tektites (Irghizite, Libyan Desert Glass, Moldavite, Muong Nong, Thailandite) and impact glasses (from the Dellen, El’gygytgyn, Lonar, Mien, Mistastin, and Popigai impact structures) were analyzed in four size fractions of (0-25, 25-63, 93-125 and 125- 250 µm) from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5-19 µm in bi-directional reflectance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The characteristic Christiansen Feature (CF) is identified between 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3 µm (Libyan Desert Glass) and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2 µm (Dellen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Most samples show mid-infrared spectra typical of highly amorphous material, dominated by a strong Reststrahlen Band (RB) between 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 µm (Libyan Desert Glass) and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3 µm (Dellen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Even substantial amounts of mineral fragments hardly affect this general band shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Comparisons of the SiO2 content representing the felsic/mafic composition of the samples with the CF shows felsic/intermediate glass and tektites forming a big group, and comparatively mafic samples a second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' An additional sign of a highly amorphous state is the lack of features at wavelengths longer than ~15 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The tektites and two impact glasses, Irghizite and El’gygytgyn respectively, have much weaker water features than most of the other impact glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' For the application in remote sensing, spectral features have to be correlated with compositional characteristics of the materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The dominating RB in the 7-14 µm range correlates well with the SiO2 content, the Christiansen Feature shows similar dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' To distinguish between glass and crystalline phases of the same chemical composition, a comparison between CF the SCFM index (SiO2/(SiO2+CaO+FeO+MgO)) (Walter and Salisbury, 1989) is useful, if chemical compositional data are also available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Introduction This study focuses on mid-infrared reflectance spectra of melt glasses from impact events for the application in planetary remote sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The obtained spectra will be part of a database for the ESA/JAXA BepiColombo mission to enter the Hermean orbit in 2024 (Maturilli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2008, Benkhoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010), and general remote sensing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The ESA/JAXA BepiColombo mission includes a mid-infrared spectrometer (MERTIS-Mercury Radiometer and Thermal Infrared Spectrometer) that allows for mapping the surface of Mercury in the 7-14 µm range, with a spatial resolution of ~500 m (Helbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Benkhoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Hiesinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Laboratory FTIR (Fourier Transformed Infrared) spectra of minerals and rocks, but also of synthetic analogs, have to be collected to be compared to those from MERTIS for the interpretation of the remote sensing data from the surface of Mercury (For a discussion of the comparison of bi-directional reflectance and emission data see below, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='6 Bi-directional reflectance FTIR-Analyses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The surfaces of terrestrial planets and their moons are shaped and modified by impact events throughout their lifetimes (Hörz and Cintala, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Thus, the investigation of how these processes affect the spectral properties of the rocks, which are modified or produced during the impact event is important for the interpretation of infrared data from planetary bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In particular, higher shock ranges result in amorphous phases produced in solid state transformation (such as maskelynite), or melt glass from complete melting of the target material (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Stöffler, 1966;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Ostertag, 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Koeberl, 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Dressler and Reimold, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Osinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Wünnemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Osinski and Pierrazo, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Jaret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2015a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Pickersgill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The aim of this study is to provide mid-infrared spectra in the 7-14 µm range for bulk impact melt rocks, as well as separated glasses for four size fractions (0-25, 25-63, 63-125, 125- 250 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Motivation for studying several size fractions is that a high porosity and large grain size variations are characteristic properties of surface regolith material, which affects the 4 spectral properties of the materials and thus have to be taken into account in the interpretation of remote sensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Variation in grain size cause the intensity of the characteristic Reststrahlen Bands (RB), fundamental mode absorption features in the 7-14 µm region, to loose spectral contrast with decreasing grain size, so the intensity of the bands gets weaker (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Salisbury and Eastes, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Salisbury and Wald, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Mustard and Hayes, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Ruff and Christensen, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' As a consequence, the identification of phases based on band positions my get difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Earlier studies in the visible and near-infrared indicate a strong influence of grains smaller 30 µm in sizes for the surface regolith on Mercury (Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2007), so the RB can expected to be weak in the remote sensing data of this planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Furthermore, a characteristic additional spectral feature, the Transparency Feature (TF) appears around 11 – 13 µm in the smallest grain size fractions as a result of increased volume scattering (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Salisbury, 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Such features may have been observed in earlier, ground based infrared observation of Mercury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Thus laboratory spectra showing this diagnostic band are also of high interest (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' So a series of analyses of distinct size fractions is needed to distinguish between genuine effects and effects induced by mixed grain size fractions in the natural regolith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' However, since regolith is also an intimate mixture of many phases, spectral unmixing modelling will also have to take many other grain-size fractions of other potential mineral phases into account, which will complicate the discussion of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The present work is a follow-up on an earlier study, where we analyzed Suevite impact rocks from the Nördlinger Ries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The former study shows a high degree of amorphization among samples of the Suevite impact rock that contains material in all stages of shock metamorphism (Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Here we focus mostly on impact glasses formed from quenched impacts melts as representatives of the highest shock stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In shock metamorphism, mineral phases start to change above 2 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Planar deformation features appear between 8 and 25 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In 5 addition, microscopic changes in the crystal structure of quartz and feldspar arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' At pressures between 25 -40 GPa, minerals turn into diaplectic glasses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' maskelynite) in solid-state transformation, forming of and amorphous material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Onset of melting of phases starts with feldspar at pressures from ~35 to ~45 GPa, over 60 GPa rocks completely melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Vaporization follows at pressures above 100 GPa (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Stöffler, 1966, 1971, 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Chao, 1967;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' von Engelhardt and Stöffler, 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Stöffler and Langenhorst, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' French, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Johnson, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' However, these pressures are only approximate to provide a general picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Shock pressures and therefore their effects on minerals are often heterogeneously distributed in the rocks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Hanss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Furthermore, they are based on crystalline, non-porous rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Different abundances of mineral species like quartz and feldspar could affect the exact pressure range since they show differences in shock transformation (Ostertag, 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Stöffler and Langenhorst, 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Petrology (Hörz and Cintala, 1997) but also structural features like grains size or porosity might play a role (French, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Wünnemeann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Also, variations in mineral chemical composition are also important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' For example, Ca-rich plagioclase transforms into maskelynite at lower pressures than K- and Na-rich feldspars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Ostertag, 1983, Johnson and Hörz 2003, Johnson, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In order to obtain a more comprehensive understanding of the spectral properties of highly shocked rocks, we present new mid-infrared spectra of impact glasses originating from a series of terrestrial impact craters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' These glasses are found in the impactites in crater structures, but are also derived from distant ejecta, the so-called tektites (Dressler and Reimold, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' We selected impact glasses and tektites from a series of terrestrial craters (for all details see Table 1) in the age range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='6 Ma (Lonar crater, Jourdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2011) to 119 Ma (Mien impact structure, Bottomley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The size of the respective impact craters or structures (if known) ranges from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='8 km for Lonar crater (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2011) to about 100 km for the 36 Ma old Popigai impact structure (Bottomley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Kettrup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The Canadian Mistastin Lake impact structure is represented in our study by two melt glass samples (Grieve, 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Marion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Pickersgill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 6 The composition of the basement rocks, which are the assumed source material of melts, is varied, but in most craters dominated by granodioritic and gneissic basement rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The El’gygytgyn impact structure, is an exception with the basement being dominated by rhyolitic ignimbrites (Layer 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Wittmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Raschke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The Lonar crater formed in basaltic basement material (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Irghizite from the Zhamanshin impact structure (Deino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1990) formed in a very diverse target material, consisting of schist, phyllite, clays and sandstones (Deino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Magna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Ackerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' A special type of impact glass is called tektite, which is usually very pure, homogenous SiO2-rich glass with low abundances of inclusions and typically particles display an aerodynamic shape (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Koeberl, 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' French, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Dressler and Reimold, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In our study they are represented by samples from two strewnfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The Central European strewnfield is represented by Moldavites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Thailandites and Muong Nong tektites are from the Australasian strewnfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Further strewnfields not covered in this study are the Ivory Coast and North American ones (French, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In the case of the Australasian tektites (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='77 Ma, Izett and Obradovich, 1992) the large required source crater (up to 116 km diameter) is unknown (Lee and Wei, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Son and Koeberl, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Boron isotope studies point towards marine or river sediments as source material (Chaussidon and Koeberl, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The probable source materials of Moldavites were likely surface sediments of the Ries impact site, mainly composed of sands and clays (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', von Engelhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Impact glass from the Nördlinger Ries crater is represented by two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Otting Glasbombe is glass separated from Suevite originating in the Otting site in the Ries, while the Polsingen sample is from the Red Suevite, basically a coherent melt rock (Stöffler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Morlok et al, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Similarly, the source crater for the Libyan Desert glasses is also unknown, but their origin from an impact into a sand/sandstone type material is strongly indicated by the occurrence of christobalite, meteoritic components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', chondritic ratios of Co, Cr, Fe, Ni, and Ir) and high- 7 temperature phases like baddeleyite (ZrO2) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Barnes and Underwood, 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Storzer and Koeberl, 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Rocchia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Greshake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Fröhlich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The mostly granitic and felsic petrology of the crystalline basement of most of the impact structures is quite different compared to that of Mercury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' MESSENGER data indicates that the best analogs for the surface of Mercury are basalts or ultramafic komatiites (Nittler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Stockstill-Cahill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Charlier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Maturilli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Unfortunately, naturally shocked basalts are generally rare (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Still, a spectral study of shocked felsic material is of interest for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' It gives insight into the spectral behavior of highly shocked impactites and their components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The data are also of use for remote sensing of terrestrial impact sites, where impactites with a felsic mineralogy are common (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Studies of other planetary surfaces than Mercury, can also benefit from results obtained in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' For example, granitic material occurs as fragments and clasts in lunar samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Warren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Jolliff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Shervais and McGee, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Seddio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Glotch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' (2010) observed evolved lithologies and highly silicic material (potential lunar granite) in the mid-infrared using the Diviner Lunar Radio experiment in four lunar regions including the Aristarchus crater, where Mustard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' (2012) found high-silica impact melt based on Moon Mineral Mapper data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' On the dominantly basaltic Mars granitoid or felsic materials may occur (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Ehlmann and Edwards, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Sautter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2014, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Furthermore, there are indications for felsic, granitoid material on Venus (Müller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Gilmore, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Earlier reflectance and emission studies of impact melt glass and amorphous phases formed in impacts in the mid-infrared were performed, for example, by Thomson and Schultz (2002), Gucsik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2004), Faulques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2005) and Palomba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Spectra of these samples are dominated by a broad feature in the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 µm range, with only few other features in the mid-infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2011) and Basavaiah and Chavan (2013) analyzed bulk material, from the basaltic Lonar Crater, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Jaret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2013, 2015a) analyzed 8 individual minerals using infrared microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' All results show a spectrum with a dominating feature in the ~9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='4 - 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2 µm wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Studies on experimentally shocked natural samples (anorthosite, pyroxenite, basalt and feldspar) were made by Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2002, 2003, 2007, and 2012) and Jaret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In these studies, a degradation of features (loss of intensity, band shifts with increasing shock pressure) were observed due to decreasing structural order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' A transmission study of experimentally shocked feldspars by Ostertag (1983) show similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Similar results with strong features between 9 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 µm were found by Pollack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (1973);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Crisp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (1990);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Nash and Salisbury (1991) and Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2001), who studied terrestrial glasses in basalts or obsidian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Several mid-infrared reflectance and emission studies used synthetic glasses as analogues for impact melt glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Byrnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2007), and Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2010) analyzed synthetic quartzofeldspathic glasses, finding very good correlations between the band position of characteristic dominant features and SiO2 contents or Si/O ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Dufresne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Minitti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2002) and Minitti and Hamilton (2010) obtained comparable results for synthetic glass with basaltic to intermediate composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Basilevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Moroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2010) and Morris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2000) measured glass from laser pulse experiments with Martian soil analogue JSC Mars-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The resulting spectra are dominated by a strong single feature in the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 µm wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' There are also many studies of melt glasses and highly shocked materials in the visible and near-infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Raikhlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (1987), Schulz and Mustard (2004) and Iancu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' (2011) studied terrestrial impact melt rocks, Johnson and Hörz (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (1979) and Bruckenthal and Pieters (1984) experimentally shocked feldspars and enstatite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Moroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2009) studied synthetic Martian impact melt analogs and Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' (1976) and Stockstill-Cahill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2014) analyzed synthetic lunar glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' A study by Keppler (1992) focused on synthetic silicate glasses with albite and diopside composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 9 In this study, provide mid-infrared data in the 7-14 µm range of further impact glasses which were so far not covered by mid-infrared studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Since the surfaces of many planetary bodies are covered by regolith, we have to expect intimate mixtures of grains and particles of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Grain sizes affect the spectral properties of materials, so this study covers several grain size fractions (0-25, 25-63, 63-125 and 125-250 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' A further question is the identification of spectral parameters, which help to identify glasses in remote sensing spectra, and to distinguish them from crystalline material with similar chemical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Material and Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 Samples In order to get representative spectra, we used several grams of melt glass for each of our analyzed samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Samples (Table 1) were obtained from the collections of the Institut für Planetologie (Münster), Mistastin samples were provided by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Zanetti (St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Louis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Samples of Lonar crater, Libyan Desert Glass and Irghizite were obtained from meteorite dealers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The Otting and Polsingen samples were made with material used already in Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2 Sample Preparation For the grain size fractions the bulk sample material was first ground in steel and agate mortars into fine powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The material was cleaned in acetone and dry sieved for at least one hour into four size fractions: 0-25 µm, 25-63 µm, 63-125 µm and 125-250 µm, using an automatic Retsch Tap Sieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' To remove clinging fines, the larger two fractions were again cleaned with acetone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 10 For additional analyses using optical and Scanning Electron Microscopy (SEM), we produced thick sections of representative blocks of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The sections were polished according to standard procedures for petrological thin sections, guaranteeing a very flat surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3 Optical Microscopy Overview images of the polished thick sections under normal light were obtained with a KEYENCE Digital Microscope VHX-500F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Light microscopy provides fast information about the general homogeneity, amorphous character, as well as enables first mineral identification in the samples (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='4 SEM/EDX For imaging and the chemical characterization of the samples, we used a JEOL 6610-LV Scanning electron microscope equipped with an implemented silicon drift Oxford EDX (Energy Dispersive X-Ray Spectroscopy) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Images were obtained in the Backscattered Electron modus (BSE) to enhance contrast due to chemical variations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Chemical analyses with EDX were quantified with an ASTIMEX™ standard set for major elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The calibration was confirmed by re-analyzing the standards in each session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Beam current stability was controlled for each analysis using a Faraday cup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' For measurements of the chemical composition, we routinely analyzed areas of 100 x 100 µm2 using 90 seconds integration times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' A broad beam and shorter integration times are helpful to measure volatile elements correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Results are presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='6 Bi-directional reflectance FTIR-Analyses Each size fraction was gently placed in aluminum sample cups (1 cm diameter, and 1 mm deep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The surface was flattened with a spatula following a similar procedure described by Mustard and Hayes (1997) to create a uniform surface without preferred grain orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' For the analyses in the mid-infrared from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5-19 µm, we used a Bruker Vertex 70 infrared system with a MCT detector at the IRIS (Infrared and Raman for Interplanetary Spectroscopy) laboratory at the Institut für Planetologie (Münster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Most measurements were conducted under low pressure (10-3 bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In some cases we observed pore collapse during evacuation even in repeated attempts, resulting in surface distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The probable reason for pore collapse is rapidly expanding volatiles when low pressures are reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Attempts to avoid this happening using longer evacuation times or pre-heating to get rid of adsorbed volatiles were also not successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' So we decided to analyze these sample und ambient pressure (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' This may have resulted in water and atmosphere-related features overlapping with overtone features in the spectral range below 7 µm but is unlikely to affect longer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' To ensure a high signal-to-noise ratio, we accumulated 512 scans for each size fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' For background calibration a diffuse gold standard (INFRAGOLDTM) was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In order to emulate various observational geometries of an orbiter, we obtained analyses in a variable geometry stage (Bruker A513) for the MERTIS database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The data presented here were obtained in a specular geometry of 30° incidence (i) and 30° emergence angle (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Results of further analytical geometries are not presented, since the differences are in the intensities of the spectra, while effects on band positions were not observed in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Band positions for both powder and microscope analyses of the characteristic features were obtained using Origin Pro 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The wavelength of a specific feature was determined by the position of the strongest reflectance, in the case of the CF we used the position of the lowest reflectance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 12 For the comparison with remote sensing data in the thermal infrared, emission and reflectance data can be compared using Kirchhoff’s law: ε = 1 – R (R=Reflectance, ε = Emission) (Nicodemus, 1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' However, in this study, a bi-directional, variable mirror set-up was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' This affects the conversion of the reflectance data to emissivity considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Kirchhoff’s’ Law works well for the comparison of directional emissivity and directional hemispherical reflectance (Hapke, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Salisbury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1994), for a direct comparison of directional emissivity with reflectance by using Kirchhoff’s law, the reflected light in all directions has to be collected, using a hemispherical reflectance, where a gold-coated hemisphere (Salisbury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1991a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Thomson and Salisbury, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' This makes direct quantitative comparisons between the data obtained in bi-directional mode with surface emission data difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' For example, anomalous features heave been observed for the CF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' (Salisbury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1991b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Salisbury, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Christiansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' However, similarity in band positions and band shapes, as well as the low amount of sample materials needed for the bi-directional analyses (which is important for rare or pure phases) makes this method useful at least for qualitative studies, when these caveats are kept in mind for direct comparisons with remote sensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The environmental conditions on the surface of Mercury also have to be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Temperatures on the surface can reach up to 700K, at a very low ambient pressure below a pico- bar (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Benkhoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In contrast, analyses for this study were conducted at low to ambient pressure, and room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Emissivity studies of olivine, pyroxene and feldspar under conditions similar to the surface of Mercury or the Moon (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Donaldson Hanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Helbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2009 and 2013) show significant shifts in band positions and also a decrease in spectral contrast for the RB and TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' This also affects direct comparisons between bidirectional reflectance spectra recalculated to emissivity with the remote sensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The spectral range of interest for the MERTIS database is from 7-14 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' We present powder spectra from 7-19 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 3a-c), as features of interest can appear at longer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The signal of the detector used in our study is less effective at wavelengths above 19 µm, resulting in a low signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Spectra are presented in reflectance, from 0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' For the spectral range 13 from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5-7 µm, where the bands for water are located, we present representative spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' For characterization purposes, we used reference spectra from the Arizona State University Thermal Emission laboratory (Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2000) and the Johns Hopkins ASTER laboratory (Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Characterizing the Samples using Optical Microscopy and SEM/EDX Focus of our chemical analyses was to obtain the composition of the pure glass and that of inclusions separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Mistastin Melt 2 shows low totals below 98 wt% (Table 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Similar low totals were observed for Ries glasses in the present and earlier studies (Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' These low totals are probably results of high sample porosities and volatile contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Chemical analyses usually confirm earlier results (for references see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' For better identification, low analytical totals of mineral inclusion were normalized to 100wt% (Table 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The Thailandite and Moldavite tektite samples have all a transparent appearance in optical microscopy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The Thailandite tektite only shows a small inclusion (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 mm) in the otherwise nearly pure glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The inclusion consists of rutile and/or SiO2 polymorphs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 and Table 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' SEM images also show very homogeneous material (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Optical images of Moldavite reveal dendritic crystal aggregates (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Optical images of the Muong Nong sample (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1) show abundant layers of small vesicles, which are typical for this type of tektite (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Koeberl, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Son and Koeberl, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Backscattered Electron (BSE) SEM images confirm the abundant small, empty vesicles (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The Libyan Desert Glass has empty vesicles in an otherwise very homogeneous, colorless glass, with only few small inclusions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1) of a crystalline SiO2-polymorph, probably cristobalite, which is typical for this material (Table 2b) 14 (Barnes and Underwood, 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Fröhlich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' SEM images of the glassy part also show very homogeneous material (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The El’gygytgyn melt glass shows Schlieren-like structures as well as few mineral inclusions in the optical images and in the same SEM pictures, embedded in a homogenous, transparent glass (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Inclusions of quartz, magnetite, and ilmenite were observed by Gurov and Koeberl (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The Irghizite impact glass has a brown color, and shows abundant (Koeberl and Fredriksson, 1986) empty vesicles in the optical and SEM images, but no larger inclusions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Popigai melt glass also shows a brown color in the optical images, with abundant vesicles and also some mineral inclusions in the glassy matrix (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' This was also observed by Whitehead et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2002) and Kettrup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=',(2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The inclusions were identified as a SiO2-polmorph with SEM/EDX, easily distinguishable from the glass in the SEM image by the blocky appearance (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2, Table 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Kettrup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2003) identified Lechatelierite e and coesite in Popigai glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The Mien impact melt glass is very heterogeneous, with abundant inclusions in the glassy matrix (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The inclusions were characterized as SiO2-polymorphs and pyroxene by SEM/EDX (Table 2b, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' This confirms the findings of Maerz (1979), who observed abundant quartz and pyroxene mineral fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Vesicles also show signs of remnant fillings (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Of the Dellen sample, only powdered material (size >250 µm) was available, so detailed information about the petrographical context is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Deutsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' (1992) reported a glassy matrix with crystallites e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', pyroxene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' This is consistent with bulk chemical analyses obtained by SEM/EDX in this study (Tab2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The samples from the Mistastin lake impact structure have considerable variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Mistastin melt 1 shows abundant fragments and vesicles, embedded in a brownish, but translucent glass (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Fragments identified by EDX were SiO2-polymorphs (Tab 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Mistastin melt 2 has a dense, black obsidian-like groundmass with abundant crystallites (also compare Grieve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1975) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The SEM images also show crystallites in a fine-grained matrix (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Further 15 phases observed were SiO2-polymorphs, probably quartz (Table 2b) (also see Marion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010) The Lonar crater sample is highly vesicular, and the glassy material contains abundant mineral inclusions, ranging from fragments to clusters of small phases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Identified by SEM/EDX were SiO2-polymorphs, pyroxene, and ilmenite (Table 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' This confirms findings of Kieffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (1976), Osae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' (2005), and Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Impact melt glass from the Lonar crater has average MgO (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='06wt %) and CaO (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='53wt %) (Table 2a) contents slightly outside the range observed for melt rocks and breccia in Osae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2005 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='32 wt% MgO and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='81 wt% CaO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Given the heterogeneity and weathered nature of the material, such divergence is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Bi-Directional Reflectance FTIR All impact melt glasses and rocks show similar spectral features in the 7-14 µm wavelength range (Table 3, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3a-c): A strong RB dominates between 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3 µm (Features below 7 µm will be discussed separately in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The Christiansen Feature (CF), a reflectance minimum, varies from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3 µm (Desert Glass) to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2 µm (Dellen)(Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In many samples, the CF is located at slightly higher wavelengths in the finest size fraction (0-25 µm) when compared to the coarser ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The TF usually only occurs in the finest size fraction (0-25 µm) (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' To aid discussion and interpretation, groupings among the spectra and samples, in addition to groupings based on chemistry and their source material (tektites, impact glass), were identified by comparing CF to the SiO2 content (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The position of the CF is well correlated with the SiO2 content of the material (Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2002) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In order to avoid artefacts caused by low analytical totals resulting from high porosity in a few samples, we compare the SiO2 content from EDX analyses normalized to 100 wt% totals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Most of the impact glasses form a group consisting of the felsic/intermediate samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' This group overlaps with the tektites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Glasses from Mistastin and Lonar, while chemically still at the 16 lower end of the intermediate range (>52 wt% SiO2, Le Maitre, 2002), form the (in relative terms) ‘mafic’ group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Silicate glasses show an asymmetric stretching of the Si-O bonds in the 8 – 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 µm region, overlapping with an asymmetric (Si, Al)-O stretching mode near 9 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The specific bands are difficult to distinguish in glass, which results in the characteristic broad band for amorphous silicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Further Al-O, Si-O-Si and O-Si-O vibrations occur at longer wavelengths over 11 µm (McMillan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2004, Dufresne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2009, Speck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In more detail, the group of tektites (Table 3, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3a) shows similar spectral characteristics, a CF between 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='7 µm, a strong RB from 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2 µm, and a weak TF between 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='62 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The spectra are very smooth, without significant spectral features of crystalline species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The intensities of the strongest RB is also very similar at ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 Reflectance (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Also, there are no significant features at longer wavelengths, except for a weak feature between 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='8 and 13 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The bulk of melt glasses and rocks of the felsic or intermediate groups show a greater spectral variation that also overlaps with those of tektites (Table 3, Fig3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' El’gygytgyn, Irghizite, Popigai, Mien, Libyan Desert Glass, Otting, and Polsingen, show simple spectra with shapes similar to tektites (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The spectrum of Libyan Desert Glass is flat at the longer wavelengths as are the spectra of tektites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Most samples show a shoulder between 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 – 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='4 µm, especially pronounced in the Libyan Desert Glass (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3a-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Such shoulders or bands are characteristic for silicate glasses with intermediate composition and related to asymmetric Si-O-Si bridge vibrations (McMillan and Pirou, 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Dufresne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Positions for the characteristic spectral features range from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 µm for the CF, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 µm for the strong RB, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 µm for the transparency feature (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Intensities show more variation with the strong RB between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 Reflectance (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Exceptions are the Libyan Desert Glass with the highest intensity in this study, and the Polsingen sample with the lowest intensity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Libyan Desert Glass is the endmember with these strong RB features occurring at shorter wavelengths, the band positions are also comparable to those of 17 synthetic SiO2 glass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Faulques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The features in the Ries samples are seen at longer wavelengths (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Furthermore, in most samples in this group weak features are observed in the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='8 µm range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Red Suevite from Polsingen shows several smaller bands between 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='8 µm and also exhibits stronger bands at longer wavelengths of 17-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='7 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3b), which are probably caused by the crystalline inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' These are typical bands of fragments with granite (or similar) composition from unmelted crystalline basement rocks (Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2009, Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In most cases, at longer wavelengths, there are only weak additional features comparable to the tektites (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 3), probably result of Al-O, Si-O-Si and O-Si-O vibrations (McMillan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The samples from Dellen are an outlier (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The CF is at 8-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2 µm, and the TF at 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 µm (Table 3b, Fig3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The spectra shows no clear main RB, instead it features a ‘twin peak’, band from 10-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3 µm which are pyroxene and plagioclase features from fragments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2009, Hamilton et al, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The strong additional features at longer wavelengths (>17 µm) in Dellen and Polsingen are indicative of weathering phases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' clay;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2016) (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Samples from Mistastin and the Lonar craters form the (relatively) mafic group (Table 3, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The CF falls between 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 µm, and the TF is between 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='8 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The position of the dominating RB feature ranges from 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2 to 10 µm, and the intensities of the strongest RB are below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 Reflectance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' However, a second feature observed in Lonar at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='6 µm (Table 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3b) could be a shoulder or feature from unshocked or moderately shocked feldspar (Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Also, comparatively strong features at longer wavelengths (>17 µm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Table 3) hint at weathering phases (Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2016) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Table 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In a similar way, the shoulder of the Mistastin Melt 2 spectrum at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='6-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='7 µm and the features from 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='6 µm (Table 3) can be explained by the abundant feldspar crystals occurring together with the melt glass, and possibly 18 weathering phases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1,2) (Grieve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Marion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In general, the strength of the RB feature between 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3 µm in all spectra shows of this study the dominance of amorphous material in the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 Water Bands The detected water bands were normalized to unity in the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5-7 µm regions to allow for better comparison of the relative intensity of the water feature (Table 3, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Although the position of the water band is very similar in all the samples (it falls between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='7 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 µm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Faulques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2001), the intensities of the feature vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' However, several of the samples have been analyzed under ambient pressure, so they could be influenced by atmospheric water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Tektites, as well as Irghizites and El’gygytgyn show very shallow water bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Most other samples exhibit broader bands, indicating adsorbed water in addition to OH groups as part of the mineral structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Libyan Desert Glass has a much sharper feature, indicating a very ‘dry’ sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Discussion Results for the strong, dominating RB in impact glasses and tektites in this study (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3 µm, Table 3) are similar to those obtained in earlier studies, which also exhibit the strong RB band in the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 µm range (Thomson and Schultz, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Gucsik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Faulques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Palomba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In addition, our results are also comparable to terrestrial glasses (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='0-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 µm) Pollack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (1973);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Crisp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (1990);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Nash and Salisbury (1991) and Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Series of synthetic glasses based on terrestrial rock compositions (Byrnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Minitti 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' DuFresne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Minitti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010) show a similar range for this feature from 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='4 µm (for comparisons with the CF see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 19 The spectrum of Muong Nong in our study is very similar in band shape and position compared with results obtained by Gucksik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' (2004): the strongest RB feature is at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2 µm, compared to an average of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2 µm in this study (Table 3, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' An Indochinite spectrum with a strong RB at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 µm (Faulques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2001) shows similarity to both Muong Nong (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2 µm) and Thailandite (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Another feature near 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 µm (Faulques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2001) is found nearby in the results of our study (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The dominating RB at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 µm for Moldavite and a weaker band at 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='8 µm is close to those in Faulques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' (2001) for the same tektite (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='0 µm and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='8 µm) (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' A minor band at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 µm (Faulques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2001) is lacking in our results (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The range of results for the main RB in tektites are also comparable to those for synthetic quartzofeldspathic glass with similar composition in Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2010), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2 µm for this study (Table 3) compared to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The weak water bands are in good agreement with low water contents (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='002-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='03 wt%) reported in tektites by Beran and Koeberl, 1997 and Dressler and Reimold, 2001) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Other mid-infrared tektite studies are mostly transmission spectra, which cannot be directly compared to our reflectance data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' However, the typical shape dominated by a RB feature near 9 µm in tektites is similar to that observed in this study (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Fröhlich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2013, Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' When compared to synthetic glasses with similar composition (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010), the range in the position for the main RB for the felsic/intermediate group in our study is comparable (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Minor features in the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='8 µm and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='8 µm ranges observed for most member of this group can be attributed to crystalline quartz inclusions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Among the felsic and intermediate impact glasses, end member Libyan Desert Glass has very similar features like FTIR analyses by Faulques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2001), which show RB at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 µm, and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='8 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In contrast, the spectra of Gucsik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2004) has only one RB at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='6 µm, and another weaker feature at 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='4 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' These differences are indicative of the compositional heterogeneity of the Libyan Desert Glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 20 In the felsic and intermediate group, Irghizite and El’gygytgyn samples have weak water bands like previously seen in the tektites (Beran and Koeberl, 1997) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' This indicates very high temperatures during their formation that dehydrated the material, or low abundance of water in the starting material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The stronger water bands of the remaining samples in this group could either point toward higher water contents in the target material, or the influence of weathering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' This was also observed for impactites from the Nördlinger Ries crater, where over 2 wt % H2O was observed in some glasses (Vennemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Similarly, elevated water (or volatile) contents of > 1 wt% were found in melt rock and glass from Dellen (Deutsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1992), Mien (Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1997) and Popigai (Kettrup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In addition, the elevated water content of the Libyan Desert Glass compared with tektites was also observed by Faulques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Results from Dellen samples are an outlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Based on its SiO2 content, the RB and CF should be at shorter wavelengths (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' However, the positions are not surprising given the number of crystalline inclusions in the glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' For example, the observed high abundance of pyroxenes (Table 2) could have moved the band position of the feature to longer wavelengths (Salisbury, 1991a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Deutsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In the more ‘mafic’ group (compared to the previously discussed felsic/intermediate group), the spectrum of the Lonar samples exhibit the strongest RB at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2 µm (Table 3), similar to findings of Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2011), which have the RB at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='4-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 µm for samples showing transformation to maskelynite (class 2) to complete melting (class 5) (Kieffer et all, 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2011) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Jaret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' (2015a) identified a similar CF position at ~8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='0 µm (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='0-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 µm in this study, Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' On the other hand, in-situ analyses of completely amorphous material from Lonar have the main feature at 10-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='4 µm (depending on grain orientation), while a feature at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3 µm occurs in spots which retained some crystallinity (Jaret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2013, 2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Differences between the spectra can be explained with the highly heterogeneous character of the materials resulting in high contents of fragments (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1, 2) (Kieffer, 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Osae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The 21 normalized water feature of the Lonar material is comparatively weak, the intensity is located between that of most samples and the tektites (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The two Mistastin melts show slightly different spectra (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3c), likely due to the degree of crystallinity in the different samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Mistastin Melt 2 has additional features (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='6-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='7 µm, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1- 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='6 µm (Table 3) explained by the abundant crystals within the melt glass (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1,2) (Grieve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Marion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Mistastin Melt 1, on the other hand, has a spectrum typical for entirely amorphous materials (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The position of the strong RB (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='6-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='7 µm, Table 3) is similar to that of synthetic glasses (~9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='8 µm, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The water bands are comparable to those for the other studied melt glasses (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Identification of Glass Glasses are characterized in the context of this study by a broad main RB in the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3 µm range (Table 3), in contrast to the usually more feature-rich crystalline felsic and intermediate rocks (Salisbury, 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' A comparison of this dominant RB with SiO2 content shows similar characteristics like the comparison of the CF and the SiO2 content (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' With increasing SiO2 content, the band position of the RB moves to shorter wavelengths (also compare Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' However, in some cases remote sensing data only provides spectra with weak RB features e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' in many ground based telescope observation in the mid-infrared of Mercury (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Tyler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1988, Emery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1998, Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2001, Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2002, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' So, additional spectral features to identify glassy material are of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' If the CF as reflectance minimum (or emission maximum) is obtained in such observations, it is comparatively easy to identify in remote-sensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In a direct comparison between CF and SiO2-content (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5), the results of our study plot on a slope very similar to that observed for a series of powdered crystalline plutonic rocks in earlier works (Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2002), with only the exception of the Dellen and the Libyan Desert Glass data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Thus, on the basis of this type of diagram, it would be difficult to distinguish melt glass from crystalline material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The 22 independence of the CF from degree of crystallinity was already observed e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' for feldspar by Nash and Salisbury, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' An alternative is the comparison between the CF and the SCFM (SiO2, CaO, FeO, MgO) index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The abundance of divalent cations Ca, Fe, and Mg affects the depolymerization of the silicate tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The SCFM index is calculated from the oxides of Si, Ca, Fe, and Mg: SiO2/(SiO2+CaO+FeO+MgO) (Walter and Salisbury, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The results (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='7) show essentially a similar grouping as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' However, impact melt glasses differ slightly from the trend line for powdered crystalline rocks with comparable chemical composition using the SCFM index (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='7), with most results plotting below the trend line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The SCFM is sensitive to the degree of polymerization of the rocks (Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2002), while glasses have a low degree of ordering with only short range structures present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' This could explain the divergence, and so SCFM values plotting below the line for crystalline rocks are possibly indicative of melt glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' However, in a remote sensing situation the divergence would require additional chemical data for the observed material in order to differentiate between glass and crystalline material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Chemical data for surface regions on Mercury was calculated form data obtained by the X-Ray Spectrometer (XRS) and the Gamma-Ray Spectrometer (GRS) on the MESSENGER probe that orbited Mercury (Charlier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Stockstill-Cahill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' vander Kaaden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Peplowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The BepiColombo mission will include MGNS (Mercury Gamma-ray and Neutron Spectrometer) and MIXS (Mercury Imaging X-ray Spectrometer) which will provide chemical data in direct alignment with the mid-infrared data from MERTIS (Benkhoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' A comparison with micro-FTIR studies of synthetic glass of felsic, intermediate, and basaltic composition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='6) (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2009) shows a difference between the natural impact melt glass and the synthetic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Since the general chemistry of the natural and synthetic glasses is comparable, effects of different analytical techniques or sample type cannot be ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' (2002) observed differences between analyses of powdered and solid rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' On the other 23 hand, Klima and Pieters (2006) observed no differences regarding band positions between these types of material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Additional spectral information in the near-infrared could also help to distinguish between crystalline and glassy material (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Gaffey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In the case of remote sensing of Mercury, such data is also available from the MESSENGER mission, and will also be provided by BepiColombo with the Visible Infrared Hyperspectral Imager Channel (VIHI) on the Spectrometer and Imagers for BepiColombo Integrated Observatory System (SIMBIO-SYS) instrument suite (Benkhoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Application to MERTIS and Remote Sensing of Mercury Mid-infrared data from Mercury is rare due to difficulties for ground-based observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Spectral studies by ground- and airplane based telescopes in the mid- infrared were made by Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 1994, 2000, 2002, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Emery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Sprague and Roush 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Donaldson-Hanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Due to observational limitations, they all cover large surface areas from at least 104 - 106 km2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The dominating mineral phases identified in these studies are mainly plagioclase with minor pyroxene (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Most spectra from Mercury show only weak features and low spectral contrast and probably have a low signal to noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Potential CF are visible from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='7 – 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='7 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The CF observed in this study overlap with this range at shorter wavelengths, with the glasses from the ‘mafic’ group, Dellen and some from the intermediate group falling into the range observed for the CF on Mercury (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The TF for the observations of Mercury is between 12 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='7 µm, only partially overlapping with the TF in our study (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 – 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Only Dellen, Mistastin and Lonar samples have CF in this region (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3) (Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 1994, 2000, 2002, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Emery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Sprague and Roush 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Donaldson-Hanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' RB are difficult to identify in the surface spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Candidates are found mostly at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 µm (Emery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 24 Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Donaldson-Hanna 2007) would overlap with most of the strong RB in the intermediate and mafic group (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Since these areas observed so far are from vast surface regions, the spectral features of distinct regions with characteristic compositions are integrated into one spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' This makes it difficult to discuss the results in a wider context, such as smaller surface features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Here the high spatial resolution of MERTIS of ~500 m (Benkhoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010) will allow to resolve smaller structures such as the hollows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' However, regolith is a mixture of many phases, so glass would be only one of several mineral/glass components and size fractions, which have to be taken into account in spectral deconvolution modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Also, the conversion problems (See 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='6 Bi-directional reflectance FTIR- Analyses) of bi-directional reflectance for comparison with emissivity data will have to be taken into account for direct comparisons of laboratory and remote sensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Summary and Conclusions We have characterized and analyzed mid-infrared spectra of four size fractions (0-25, 25-63, 93-125 and 125-250 µm) from 14 impact melt glass samples, covering the compositional range from highly felsic (Libyan Desert Glass) to mafic/basaltic (Lonar crater melt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Most samples show mid-infrared spectra typical of highly amorphous material, dominated by a strong Reststrahlen Band between 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3 µm (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Even substantial amounts of mineral fragments hardly affect this general band shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' However, crystallization from the melt, such as in one of our two Mistastin samples (Mist Melt 2), can result in significant differences in the spectra between compositionally similar samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 25 An additional sign of a highly amorphous state is the lack of features at wavelengths longer than ~15 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3a-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Tektites have much weaker water features than most of the other impact melt glasses, with the exception of samples from Irghizite and El’gygytgyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' For the application in remote sensing, spectral features have to be correlated with compositional characteristics of the materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' A convenient method to correlate compositional and spectral features is to use the prominent and in spectra with low signal/noise ratio easily recognizable Christiansen Feature and compare it to compositional parameters like the SCFM index (Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The comparison shows some differences, which could help distinguish glass from crystalline material in remote sensing, using chemical compositional data from further instruments onboard of BepiColombo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' A comparison between the laboratory spectra and mid-infrared ground-observations of Mercury shows similarities in band positions between various features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' However, there is no direct ‘fit’, the occurrence of glasses similar to those analyzed in our study on the surface of Mercury has to be confirmed by more detailed studies including a wider range of phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Acknowledgements This work is supported by the DLR funding 50 QW 1302 in the framework of the BepiColombo mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Also many thanks to Alexander Deutsch (Münster) and Mike Zanetti (St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Louis) for providing the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 26 References Ackerman L.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Spray J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' (2002) Mineralogy and petrology of melt rocks from the Popigai impact structure, Siberia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Meteoritics & Planetary Science 37, 623-647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Wittmann A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Goderis S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Claeys P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Vanhaecke F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Deutsch A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Adolph L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=" (2013) Petrology of impactites from El'gygytgyn crater: Breccias in ICDP-drill core 1C, glassy impact melt rocks and spherules." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Meteoritics & Planetary Science 48, 1199-1235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 41 Wünnemann K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Collins G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Osinski G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' (2008) Numerical modelling of impact melt production in porous rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Earth and Planetary Science Letters 269, 530-539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Wright S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Christensen P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Sharp T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' (2011) Laboratory thermal emission spectroscopy of shocked basalt from Lonar Crater, India, and implications for Mars orbital and sample data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Journal of Geophysical Research 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Wyatt M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Christensen P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Taylor L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' (2001) Analysis of terrestrial and Martian volcanic compositions using thermal emission spectroscopy: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Determination of mineralogy, chemistry, and classification strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Journal of Geophysical Research, Volume 106, Issue E7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 14711- 14732 Wünnemann K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Collins G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', Osinski G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' (2008) Numerical modelling of impact melt production in porous rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Earth and Planetary Science Letters 269, 530-539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 42 Figure Captions Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Optical images of polished sections from the studied samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Tektites Thailandite, Muong Nong and Moldavite can be recognized by their transparent appearance and very low content of inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Most other melt glasses show higher abundances of crystalline material e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Popigai, Mien), while Libyan Desert glass has an appearance similar to the tektites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Samples like Lonar or Mistastin Melt 2 have very high contents in inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' SEM/BSE images of polished sections from the samples analyzed in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The findings of the optical microscopy are mainly confirmed, high contents of vesicles are visible in the Muong Nong, Irghizite and Lonar samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Mid-infrared bi-directional reflectance spectra of impact melt and glass samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Band positions in µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' In reflectance (0-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Blue: 0-25 µm, Pink: 25-63 µm, Red: 63-125 µm, Brown: 125-250 µm (in µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' (a) Tektites, (b) Felsic/intermediate Samples, (c) Mafic/basaltic samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Spectral range from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 to 7 µm for each sample (always size fraction 125-250µm, normalized on unity), which shows the water features at ~3 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Spectra are offset along the y- axis for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Comparison of SiO2 concentration (in wt%) in the samples with the position of the Christiansen Feature (in µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Dotted line: Crystalline felsic and intermediate rocks from Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Most samples from this study form a group of felsic/intermediate material (including the tektites), with a small group of near-mafic samples (Mistastin, Lonar) forming the ‘mafic’ group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Comparison of SiO2 concentration (in wt%) in the samples with the position of the strongest Reststrahlen Band (in µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The dotted line is the trend line for the synthetic glass with intermediate to felsic composition (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Most results for the glasses differ from the results for synthetic glass (DuFresne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 43 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Comparison of the SCFM-index (SiO2/(SiO2+CaO+FeO+MgO) (Walter and Salisbury, 1989, Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2002) with the position of the Christiansen Feature (in µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Dotted line: Crystalline felsic and intermediate rocks from Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' The results for the impact glass mostly plot below the trend line for crystalline material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 1 Source Sample Age (Ma) Size crater (km) Source rock Lit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Australasian Strewnfield Muong Nong 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='8 90 116?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Marine/River sediments?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' [1,2,3,4] Australasian Strewnfield Thailandite/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='8 90 116?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Marine/River sediments?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' [1,2,3,4] Nördlinger Ries Crater Moldavite 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3 26 Sediments, sands, clays [5,6,7,8,9] Otting Polsingen 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3 26 Granite, Gneiss, Amphibolite [10] El’gygytgyn Impact Structure El’gygytgyn 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='58 18 Rhyodacitic/Rhyolitic Ignimbrite Basalt, Andesite [11,12,13] Zhamanshin Impact Structure Irghizite 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3 Quartz sericite schist, phyllite , clays and locally sandstones, Ultrabasic Intrusion [6,14,15, 16,17] Popigai Impact structure Popigai 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='7 100 Gneiss [18,19,20] Mien Impact Structure Mien 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='7 9 Granite Gneiss Amphibole [21,22] ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Libyan Desert Glass 29 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Sands/Sandstone [5,23] Dellen Impact Structure Dellen 89 20 Granodiorite Gneiss [21,24] Mistastin Lake Impact Structure Mistastin 36 28 Granodiorite, Anorthosite, Mangerite [25,26,27,28] Lonar Lake Crater Lonar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='57 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='8 Basalt [29,30,31] Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Overview of the samples and their sources used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Age in million years (Ma), Crater size in km, Basement rock: petrology of probably source material for melts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Lit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' : literature sources for data: [1] French 1998, [2] Chaussidon and Koeberl 1995, [3] Izett and Obradovich 1992, [4] Lee and Wei 2000, [5] Magna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2011, [6] Koeberl and Fredriksson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1986, [7] Laurenzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2003, [8] Buchner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2003, [9] von Engelhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2005, [10] Stöffler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2013, [11] Layer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2000, [12] Raschke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2014, [13] Gurov and Koeberl 2004, [14] Koeberl , 1986,[15] Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1979, [16] Ackerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2015, [17] Deino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1990, [18] Bottomley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1997, [19] Withehead et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2002, [20] Kettrup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2003, [21] Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1997 [22] Bottomley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1978, [23] Fröhlich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2013, [24] Deutsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=',1992, [25] Marion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2010, [26] Grieve 1975, [27] Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 1977, [28] Pickersgill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2015, [29] Osae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2005, [30] Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2011, [31] Jourdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' 2 Table 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Chemical Composition of the impact melt rocks and glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' SEM/EDX data, in wt%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Results for Otting and Polsingen are from Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='=Standard Deviation (1Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Mist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='=Mistastin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Elgy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='= El’gygytgyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Muong Nong s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Thai landite s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Molda vite s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Elgy Irghizite s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Popigai s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Mien Desert Glass s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Otting s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Polsingen s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Dellen s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Na2O 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='39 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='24 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='43 ±0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='83 Table 2 cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Mist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Melt 1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Mist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' Melt 2 Lonar s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='d.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='01 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='22 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='59 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='74 Table 3 cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Moldavite ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Moung Nong ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Thailandite ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Libyan Desert Glass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='3 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Popigai ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='El’gygytgyn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Mien ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Irghizite ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Mistastin Melt 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Mistastin Melt 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Lonar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Figure 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='2mm2 mm2mm2mm1mm2mm2mm2mm1mm2 mmMoldavite ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='100 µm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='1000 µm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Moung Nong ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='200 µm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Thailandite ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='200 µm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Libyan Desert Glass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='200 µm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Mistastin Melt2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='200 µm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Mien ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Lonar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='200 µm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Popigai ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='500 µm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='200 µm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='El’gygytgyn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='400 µm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Irghizite ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='800 µm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Mistastin Melt 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='Figure 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='8 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='05 Popigai Reflectance Reflectance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='10 Libyan Desert Glass 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='25 Reflectance 0 25 µm 125 250 µm 63 125 µm 25 63 µm 1400 1200 1000 800 600 1400 1200 1000 800 600 Band Position in cm 1 Band Position in cm 1 Otting 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,10 B A El’gygytgyn Reflectance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='10 Irghizite 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,10 0,11 0,12 0,13 0,14 0,15 B A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='02 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='15 Reflectance Figure 3 Mistastin Melt 2 Reflectance 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0,00 0,01 0,02 0,03 0,04 0,05 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='05 Reflectance 1400 1200 1000 800 600 1400 1200 1000 800 600 Band Position in cm 1 Band Position in cm 1 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,10 B A Reflectance Mistastin Melt 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='10 Figure 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E3T4oBgHgl3EQfXAr0/content/2301.04476v1.pdf'} +page_content='0 3.' 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b/INE5T4oBgHgl3EQfWw9u/content/tmp_files/load_file.txt @@ -0,0 +1,3026 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf,len=3025 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='05561v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='NT] 13 Jan 2023 LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY CHRISTOPH AISTLEITNER, ISTV´AN BERKES AND ROBERT TICHY Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In this paper we present the theory of lacunary trigonometric sums and lacunary sums of dilated functions, from the origins of the subject up to re- cent developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We describe the connections with mathematical topics such as equidistribution and discrepancy, metric number theory, normality, pseudo- randomness, Diophantine equations, and the subsequence principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In the final section of the paper we prove new results which provide necessary and sufficient conditions for the central limit theorem for subsequences, in the spirit of Nikishin’s resonance theorem for convergence systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' More precisely, we characterize those sequences of random variables which allow to extract a subsequence satisfying a strong form of the central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Uniform distribution and discrepancy 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Arithmetic effects: Diophantine equations and sums of common divisors 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The central limit theorem for lacunary sequences 12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The law of the iterated logarithm for lacunary sequences 16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Normality and pseudorandomness 20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Random sequences 24 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The subsequence principle 28 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' New results: Exact criteria for the central limit theorem for subsequences 36 Acknowledgments 49 References 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Introduction The word “lacunary” has its origin in the Latin lacuna (ditch, gap), which is a diminutive form of lacus (lake).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Accordingly, a lacunary sequence is a sequence with gaps, and a lacunary trigonometric sum is a sum of trigonometric functions with gaps between the frequencies of consecutive summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The origin of the theory of lacunary sums might lie in Weierstrass’ famous example of a continuous, nowhere differentiable function (1872).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Since then the subject has evolved into many very different directions, reflecting for example the emergence of modern measure theory and axiomatic probability theory in the early twentieth century, profound 1 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY developments in harmonic analysis and Diophantine approximation, the establish- ment of ergodic theory as one of the key instruments of number theory, or the interest in notions of pseudo-randomness which are associated with the evolution of theoretical computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Throughout this paper we will be concerned with convergence/divergence properties of infinite trigonometric series ∞ � k=1 ck cos(2πnkx) or ∞ � k=1 ck sin(2πnkx), as well as with the asymptotic order and the distributional behavior of finite trigono- metric sums N � k=1 ck cos(2πnkx) or N � k=1 ck sin(2πnkx) (the latter often in the simple case where ck ≡ 1), and with their generalizations ∞ � k=1 ckf(nkx) and N � k=1 ckf(nkx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here (ck)k≥1 is a sequence of coefficients, and (nk)k≥1 is a sequence of positive in- tegers (typically increasing), which satisfies some gap property such as the classical Hadamard gap condition nk+1 nk > q > 1, k ≥ 1, or the “large gap condition” (also called “super-lacunarity property”) nk+1 nk → ∞, k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Furthermore, f is a 1-periodic function which is usually assumed to satisfy some reg- ularity properties (such as being of bounded variation, being Lipschitz-continuous, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' ), and which for simplicity is usually assumed to be centered such that � 1 0 f(x) dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Early appearances of such lacunary sums include the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Sums of the form �N k=1 f(2kx), where f is an indicator function of a dyadic sub-interval of [0, 1], extended periodically with period 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Borel used such sums in 1909 to show that almost all reals are “normal”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' more on this topic is contained in Section 6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Uniform distribution of sequences ({nkx})k≥1 in Weyl’s seminar paper of 1916;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' more on this in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here and throughout the paper, we write {·} for the fractional part function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Kolmogorov’s theorem on the almost everywhere convergence of lacunary trigonometric series if the sequence of coefficients is square-summable (1924), LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 3 a result related to his Three Series Theorem for the almost sure conver- gence of series of independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Later it turned out that Kolmogorov’s convergence theorem for trigonometric series actually remains true without any gap condition whatsoever, a result which was widely be- lieved to be “too good to be true” before being established by Carleson [77] in 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' More on this in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Foundational work on the distribution of normalized lacunary trigonometric sums, in particular the central limit theorems of Kac (1946) and Salem and Zygmund (1947), and the laws of the iterated logarithms of Salem and Zyg- mund (1950) and of Erd˝os and G´al (1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' More on this in Sections 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A fundamental observation is that the unit interval, equipped with Borel sets and Lebesgue measure, forms a probability space, and that consequently a sequence of functions such as (cos(2πnkx))k≥1 or (f(nkx))k≥1 can be viewed as a sequence of random variables over this space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' if f is 1-periodic and if (nk)k≥1 is a sequence of positive integers then these random variables are identically distributed, but in general they are not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, under appropriate circumstances the gap condition which is imposed upon (nk)k≥1 can ensure that these random vari- ables have a low degree of stochastic dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Consequently lacunary sums often mimic the behavior of sums of independent and identically distributed random vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This viewpoint was in particular taken by Steinhaus, Kac, and Salem and Zygmund in their fundamental work on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In a particularly striking situa- tion, the dyadic functions considered by Borel actually turn out to be a version of a sequence of Bernoulli random variables which are truly stochastically independent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' accordingly, Borel’s result on the normality of almost all reals is nowadays usually read as the historically very first version of the strong law of large numbers in prob- ability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' When taking this probabilistic viewpoint, the theory of lacunary sums could be seen as a particular segment of the much wider field of the theory of weakly dependent random systems in probability theory, which is associated with notions such as mix- ing, martingales, and short-range dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, it should be noted that the precise dependence structure in a lacunary function system (f(nkx))k≥1 is con- trolled by the analytic properties of the function f, in conjunction with arithmetic properties of the sequence (nk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is precisely this interplay between probabilis- tic, analytic and arithmetic aspects which makes the theory of lacunary sums so interesting, so challenging and so rewarding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In the following sections we want to illustrate some instances of these phenomena in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Uniform distribution and discrepancy Let (xn)n≥1 be a sequence of real numbers in the unit interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Such a sequence is called uniformly distributed modulo one (in short: u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' mod 1) if (1) 1 N N � n=1 1A(xn) = λ(A) for all sub-intervals A ⊂ [0, 1] of the unit interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The word “equidistributed” is also used for this property, synonymously with “uniformly distributed modulo one”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In this definition, and in the sequel, 1 denotes an indicator function, and λ denotes Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In informal language, this definition means that a sequence is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' mod 1 if every interval A asymptotically receives its fair share of elements of the sequence, which is proportional to the length of the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that (for example as a consequence of the Glivenko–Cantelli theorem) for a sequence of independent, uniformly (0, 1)-distributed random variables (Un)n≥1 one has 1 N N � n=1 1A(Un) = λ(A) almost surely for all intervals A ⊂ [0, 1], so that in a vague sense uniform distribution of a deter- ministic sequence can be interpreted in the sense that the sequence shows “random” behavior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' more on this aspect in Section 6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Uniform distribution theory can be said to originate with Kronecker’s approximation theorem and with work of Bohl, Sierpi´nski and Weyl on the sequence ({nα})n≥1 for irrational α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, the theory only came into its own with Hermann Weyl’s [222] seminal paper of 1916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Among many other fundamental insights, Weyl realized that Definition (1), which in ear- lier work had only be read in terms of counting points in certain intervals, can be interpreted in a “functional” way and can equivalently be written as (2) lim N→∞ 1 N N � n=1 f(xn) = � 1 0 f(x) dx for all continuous functions f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This viewpoint suggests that uniformly distributed sequences can be used as quadrature points for numerical integration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' in the multi- dimensional setting and together with quantitative error estimates this observation forms the foundation of the so-called Quasi-Monte Carlo integration method, a con- cept which today forms a cornerstone of numerical methods in quantitative finance and other fields of applied mathematics (more on this below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Furthermore, Weyl realized that the indicator functions in (1) or the continuous functions in (2) could also be replaced by complex exponentials, as a consequence of the Weierstrass ap- proximation theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' thus by the famous Weyl Criterion a sequence is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' mod 1 if and only if lim N→∞ 1 N N � n=1 e2πihxn = 0 LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 5 for all fixed non-zero integers h, thereby tightly connecting uniform distribution the- ory with the theory of exponential sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For the particular sequence ({nα})n≥1 it can be easily seen from the Weyl criterion that this sequence is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' mod 1 if and only if α ̸∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, for other parametric sequences of the form ({nkα})k≥1 the situation is much more difficult, and in general it is completely impossible to determine whether for some particular value of α the sequence is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It turns out that in a metric sense the situation is quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Metric number theory arose after the clarification of the concept of real numbers, the realization that the reals drastically outnumber the integers and the rationals, and the development of modern measure theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Loosely speaking, the purpose of metric number theory is to determine properties which hold for a set of reals which is “typical” with respect to a certain measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' here “typical” means that the measure of the complement is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In the present paper the measure under consideration will always be the Lebesgue measure, and a set of reals will be considered typical if its complement has vanishing Lebesgue measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' however, metric number theory has for example also been intensively studied with respect to the Hausdorff dimension or other fractal measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Returning to Weyl’s results, what he proved in the metric setting is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For every sequence of distinct integers (nk)k≥1, the sequence ({nkα})k≥1 is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' mod 1 for (Lebesgue-) almost all reals α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In other words, even if we cannot specify the set of α’s for which uniform distribution holds, at least we know that the set of such α’s has full Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is amusing that after formulating the result, Weyl continues to write: Wenn ich nun freilich glaube, daß man den Wert solcher S¨atze, in denen eine unbestimmte Ausnahmemenge vom Maße 0 auftritt, nicht eben hoch einsch¨atzen darf, m¨ochte ich diese Behauptung hier doch kurz begr¨unden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='1 One should recall that Weyl’s paper was written in a time of intense conflict of formalists vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' constructivists (with Weyl favoring the latter ones), and only very briefly after the notion of a set of zero (Lebesgue) measure had been introduced at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Today, Weyl’s theorem is seen as one of the foundational results of metric number theory, together with the work of Borel, Koksma, Khinchin and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' While uniform distribution modulo one is a qualitative asymptotic property, it is natural that one is also interested in having a corresponding quantitative concept which applies to finite sequences (or finite truncations of infinite sequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Such 1Even if I think that the value of theorems, which contain an unspecified exceptional set of measure zero, is not particularly high, I still want to give a short justification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY a concept is the discrepancy of a sequence, which is defined by DN(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , xN) = sup A⊂[0,1] ����� 1 N N � n=1 1A(xn) − λ(A) ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here the supremum is taken over all sub-intervals A ⊂ [0, 1], and it is easy to see that an infinite sequence (xn)n≥1 is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' mod 1 if and only if the discrepancy DN(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , xN) tends to 0 as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' With a slight abuse of notation, we will write throughout the paper DN(xn) = DN(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , xN) for the discrepancy of the first N elements of an infinite sequence (xn)n≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' From a probabilistic perspective, the discrepancy is a variant of the (two-sided) Kolmogorov–Smirnov statistic, where one tests the empirical distribution of the point set x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , xN against the uniform distribution on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Without going into details, we note that DN(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , xN) can be bounded above in terms of exponential sums by the Erd˝os–Tur´an inequality, and that the error when using x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , xN as a set of quadrature points to approximate � 1 0 f(x) dx by 1 N �N n=1 f(xn) can be bounded above by Koksma’s inequality in terms of the variation of f and the discrepancy DN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for details see the monographs [96, 161], which contain all the basic information on uniform distribution theory and discrep- ancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See also [181] for a discussion of equidistribution and discrepancy from the viewpoint of analytic number theory, and [164, 165, 192] for expositions which put particular emphasis on the numerical analysis aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Weyl’s metric result from above can be written as lim N→∞ DN({nkα}) = 0 for almost all α, for any sequence (nk)k≥1 of distinct itegers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Strikingly, the precise answer to the corresponding quantitative problem is still open more than a hundred years later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is known that for every strictly increasing sequence of integers (nk)k≥1 one has (3) DN({nkα}) = O �(log N)3/2+ε √ N � for almost all α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This is a result of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Baker [38], who improved earlier results of Cassels [78] and of Erd˝os and Koksma [104] by using Carleson’s celebrated convergence theorem in the form of the Carleson–Hunt inequality [140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In his paper Baker wrote that [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' ] probably the exponent 3/2 + ε could be replaced by ε [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' ] but it turned out that this is not actually the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Instead, Berkes and Philipp [64] constructed an example of an increasing integer sequence (nk)k≥1 for which (4) lim sup N→∞ ��� �N k=1 cos(2πnkx) ��� √N log N = +∞ for almost every x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By the Erd˝os–Tur´an inequality this gives a corresponding lower bound for the dis- crepancy, which implies that the optimal exponent of the logarithmic term in an upper bound of the form (3) has to be at least 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' But the actual size of this opti- mal exponent, one of the most fundamental problems in metric discrepancy theory, LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 7 still remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that for pure cosine-sums �N k=1 cos(2πnkx) it is easily seen that one has a metric upper bound with exponent 1/2 + ε in the logarithmic term;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' this follows from the orthogonality of the trigonometric system, together with Car- leson’s inequality and the Chebyshev inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus, in connection with (4), the optimal upper bound in a metric estimate for pure cosine sums is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For sums �N k=1 f(nkx) with f being a 1-periodic function of bounded variation, the optimal exponent also is 1/2 + ε, but this is a much deeper result than the one for the pure cosine case, and was established only recently in [15, 169].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By Koksma’s inequality, an upper bound for the discrepancy implies an upper bound for sums of function values for a (fixed) function of bounded variation, but the opposite is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' So while the case of a fixed function f is solved and is an important test case for the discrepancy, the problem of the discrepancy itself (which requires a supremum over a whole class of test functions) is more involved and remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Arithmetic effects: Diophantine equations and sums of common divisors One of the most classical tools of probability theory is the calculation of expectations, variances, and higher moments of sums of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Due to trigonometric identities such as (5) cos a cos b = cos(a + b) + cos(a − b) 2 , the calculation of moments of sums of trigonometric functions (with integer frequen- cies) reduces to a counting of solutions of certain Diophantine equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' while the first and second moments � 1 0 N � k=1 cos(2πnkx) dx = 0 and � 1 0 � N � k=1 cos(2πnkx) �2 dx = N 2 are trivial and do not depend on the particular sequence (nk)k≥1 (as long as the elements of the sequence are assumed to be distinct),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' interesting arithmetic effects come into play when one has to compute higher moments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' and it can be clearly seen how the presence of a gap condition leads to a behavior of the moments which is similar to that of sums of independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' More precisely, assume that we try to calculate � 1 0 � N � k=1 cos(2πnkx) �m dx for some integer m ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By (5) this can be written as a sum 2−m � ± � 1≤k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=',km≤N 1 (±nk1 ± · · · ± nkm = 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here the first sum is meant as a sum over all positive combinations of “+” and “-” signs inside the indicator function at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Now assume that, for simplicity, we consider the particular sequence nk = 2k, k ≥ 1, which is a prototypical example 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY of a sequence satisfying the Hadamard gap condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then it is not difficult to see that ±nk1 ± · · · ± nkm = 0 is only possible if the elements of the sum cancel out in a pairwise way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' that is, after a suitable re-ordering of the indices, we need to have k1 = k2, k3 = k4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , km−1 = km, and it only remains to count how many such re-orderings are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The result is a combinatorial quantity, and it is exactly the same that arises when calculating an m-th moment of a sum of independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus the moments of the trigonometric lacunary sum converge to those of a suitable Gaussian distri- bution, which gives rise to the classical limit theorems for lacunary trigonometric sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The situation is more delicate if one only has the Hadamard gap condition nk+1/nk > q > 1 rather than exact exponential growth, and again more delicate if one considers a sum of dilated functions � f(nkx) instead of a pure trigonometric sum, but the principle described here is very powerful also in these more general situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For a long time this was the key ingredient in most of the proofs of limit theorems for lacunary sums;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see for example [103, 145, 193, 201, 214, 221].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A different method is based on the approximation of a lacunary sum by a martingale difference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' here the “almost independent” behavior is not captured by controlling the moments of the sum, but in the fact that later terms of the sum (functions with high frequency) oscillate quickly in small regions where earlier summands (functions with much lower frequency) are essentially constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As far as we can say, this method was first used in the context of lacunary sums by Berkes [53] and, indepen- dently, by Philipp and Stout [196].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We will come back to this topic in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Broadly speaking, the “almost independent” behavior of sums of dilated functions breaks down when the lacunarity condition is relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Many papers have been de- voted to this effect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see in particular [57, 59, 102, 184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In order to maintain the “almost independent” behavior of the sum, there are two natural routes to take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the one hand, one could randomize the construction of the sequence (nk)k≥1, and assume that the undesired effects disappear almost surely with respect to the under- lying probability measure – it turns out that this is a very powerful method, and we will come back to it in Section 7 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the other hand, when adapting the view- point that the “almost independence” property is expressed in the small number of solutions of certain Diophantine equations, one could try to compensate the weaker growth assumption by stronger arithmetic assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A prominent example of a class of sequences for which the latter approach has been very successfully used are the so-called Hardy–Littlewood–P´olya sequences, which consist of all the elements of the multiplicative semigroup generated by a finite set of primes, sorted in increasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' These sequences are in several ways a natural analogue of lacunary sequences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' note that the sequence (2k)k≥1 actually also falls into this framework by consisting of all elements of the semigroup generated by a single prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Such sequences gener- ated by a finite set of primes have attracted the attention of number theorists again and again, a particularly interesting instance being F¨urstenberg’s [126] paper on disjointness in ergodic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is known that Hardy–Littlewood–P´olya sequences (if generated by two or more primes) grow sub-exponentially, and the precise (only LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 9 slightly sub-exponential) growth rates are known (Tijdeman [216]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' What is more striking (and a much deeper fact) is that also the number of solutions of the rel- evant linear Diophantine equations can be bounded efficiently – this is Schmidt’s celebrated Subspace Theorem [207] in a quantitative form such as that of Evertse, Schlickewei and Schmidt [107] or Amoroso and Viada [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By a combination of the (slightly weaker) growth condition with the (strong) arithmetic information avail- able for Hardy–Littlewood–P´olya sequences, much of the machinery that is used for Hadamard lacunary sequences can be rescued for this generalized setup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see [194] as well as [19, 65, 123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We briefly come back to the case of sums of dilated functions � f(nkx) without the presence of a growth condition on (nk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We assume for simplicity that � 1 0 f(x) dx = 0, so trivially � 1 0 N � k=1 f(nkx) dx = 0, but already the calculation of the variance (6) � 1 0 � N � k=1 f(nkx) �2 dx is in general quite non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If f(x) = cos(2πx), then one can simply use the orthogonality of the trigonometric system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If f is a more general function, then one can still express f by its Fourier series, expand the square and integrate, and thus translate the problem of calculating (6) into a problem of counting the solutions of certain linear Diophantine equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' When carrying out this approach, one is naturally led to the problem of estimating a certain sum involving greatest common divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, assume that f(x) = {x}−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In this case a classical formula (first stated by Franel and first proved by Landau) asserts that � 1 0 f(mx)f(nx) dx = 1 12 (gcd(m, n))2 mn , and consequently � 1 0 � N � k=1 ({nkx} − 1/2) �2 dx = 1 12 � 1≤k,ℓ≤N (gcd(nk, nℓ))2 nknℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The sum on the right-hand side of this equation is called a GCD sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A similar identity holds for example for the Hurwitz zeta function ζ(1 − α, ·), where � 1 0 ζ(1 − α, {mx})ζ(1 − α, {nx}) dx = 2Γ(α)2 ζ(2α) (2π)2α (gcd(m, n))2α (mn)α for α > 1/2, thus leading to a GCD sum � 1≤k,ℓ≤N (gcd(nk, nℓ))2α (nknℓ)α 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY with parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If f(x) is a general 1-periodic function, then one usually does not obtain such a nice exact representation of the variance of a sum of dilated function values, but typically the variance (6) can be bounded above by a GCD sum, which together with Chebyshev’s inequality and the Borel–Cantelli lemma allows to make a statement on the almost everywhere asymptotic behavior of a sum of dilated func- tion values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This connection between sums of dilated functions and GCD sums is explained in great detail in Chapter 3 in Harman’s monograph on Metric Number Theory [136], where mainly the context of metric Diophantine approximation is treated (see also [127, 155]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Recently this connection has also led to a solution of the problem of the al- most everywhere convergence of series of dilated functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Recall that Carleson’s theorem [77] asserts that the series ∞ � k=1 ck cos(2πnkx) is almost everywhere convergent provided that � k c2 k < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is natural to ask which assumption on the sequence of coefficients (ck)k≥1 is necessary to ensure the almost everywhere convergence of the more general series (7) ∞ � k=1 ckf(nkx), under some regularity assumptions on f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Gaposhkin [129, 130] obtained some partial results, but a satisfactory understanding of the problem was only achieved very recently, when the connection with GCD sums was fully understood and optimal upper bounds for such sums were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Exploiting this connection with GCD sums, it was shown in [15, 169] that for 1-periodic f which is of bounded variation on [0, 1] the series (7) is almost everywhere convergent provided that ∞ � k=3 c2 k(log log k)γ < ∞ for some γ > 2, and this result is optimal in the sense that the same assumption with γ = 2 would not be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In [16] it was shown that for the class Cα of 1-periodic square integrable functions f with Fourier coefficients aj, bj satisfying aj = O(j−α), bj = O(j−α) for 1/2 < α < 1, a sharp criterion for the almost everywhere convergence of (7) is that (8) ∞ � k=1 c2 k exp �K(log k)1−α (log log k)α � < ∞ with a suitable K = K(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In the case of 1-periodic Lipschitz α functions f, Gaposhkin [130] proved that for α > 1/2, the series (7) converges a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' under � k c2 k < ∞ (just like in the case of Carleson’s theorem) and Berkes [60] showed LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 11 that this result is sharp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for α = 1/2 the exact analogue of Carleson’s theorem is not valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' No sharp convergence criteria exists in the case 0 < α ≤ 1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for sufficient criteria for see Gaposhkin [129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See also [4, 68, 128] for a general discussion and several further results for the convergence of series � k ckf(nkx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For general periodic f ∈ L2 the direct connection between the integral (6) and GCD sums breaks down, but upper bounds for (6) as well as for (9) � 1 0 � N � k=1 ckf(nkx) �2 dx can be given in terms of the coefficients ck, of the Fourier coefficients of f, and arithmetic functions such as d(n) = � d|n 1, σs(n) = � d|n ds, or the Erd˝os-Hooley function ∆(n) = supu∈R � d|n,u≤d≤eu 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See Koksma [156, 157], Weber [220], and Berkes and Weber [69, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A typical example (see [220]) is the bound � 1 0 �� k∈H ckf(kx) �2 dx ≤ � ∞ � ν=1 a2 ν∆(ν) � � k∈H c2 kd(k) valid for any set H of disjoint positive integers lying in some interval [er, er+1], r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here ak are the complex Fourier coefficients of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Using standard methods, such bounds lead easily to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' convergence criteria for sums � k ckf(kx), see the papers cited above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In Wintner [223] it was proved that if f is a periodic L2 function with Fourier coefficients ak, bk, then the series � k ckf(kx) converges in L2 norm for all coefficient sequences (ck)k≥1 satisfying � k c2 k < ∞ if and only if the functions defined by the Dirichlet series ∞ � k=1 akk−s, ∞ � k=1 bkk−s, are bounded and regular in the half plane ℜ(s) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' There is also a remarkable con- nection between the maximal order of magnitude of GCD sums with the order of ex- treme values of the Riemann zeta function in the critical strip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see [72, 94, 138, 209].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Naturally, estimating the integral (6) provides important information also on the asymptotic behavior of averages (10) 1 N N � k=1 f(nkx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By the Weyl equidistribution theorem, for any 1-periodic f with bounded variation in (0, 1) we have (11) lim N→∞ 1 N N � k=1 f(kx) = � 1 0 f(x) dx a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY (actually for every irrational x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Khinchin [150] conjectured that (11) holds for every 1-periodic Lebesgue integrable f as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This conjecture remained open for nearly 50 years and was finally disproved by Marstrand [178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' An example for a periodic integrable f and a sequence (nk)k≥1 of positive integers such that the averages (10) do not converge almost everywhere had already been given earlier by Erd˝os [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the other hand, Koksma [157] proved that (11) holds if f ∈ L2 and the Fourier coefficients ak, bk of f satisfy ∞ � k=1 \uf8eb \uf8ed(a2 k + b2 k) � d|k 1 d \uf8f6 \uf8f8 < ∞, and Berkes and Weber [70] proved that the last condition is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' No similarly sharp criteria are known in the case f ∈ L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For further results related to the Khinchin conjecture, see [37, 50, 70, 74, 185].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The central limit theorem for lacunary sequences Salem and Zygmund [201] proved the first central limit theorem (CLT) for lacunary trigonometric sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' More specifically, they showed that for any integer sequence (nk)k≥1 satisfying the Hadamard gap condition one has lim N→∞ λ � x ∈ (0, 1) : N � k=1 cos(2πnkx) ≤ t � N/2 � = Φ(t), where Φ denotes the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that � 1 0 � N � k=1 cos(2πnkx) �2 dx = N 2 , so the result above contains the “correct” variance for the limit distribution, exactly as it should also be expected in the truly independent case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This result has been significantly strengthened since then;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for example, Philipp and Stout [196] showed that under the Hadamard gap condition the function S(t, x) = � k≤t cos(2πnkx), considered as a stochastic process over the space ([0, 1], B[0, 1], λ), is a small per- turbation of a Wiener process, a characterization which allows to deduce many fine asymptotic results for this sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is also known that the central limit theorem for pure trigonometric lacunary sums remains valid under a slightly weaker gap condition than Hadamard’s: as Erd˝os [102] proved, it is sufficient to assume that nk+1/nk ≥ 1 + ck−α, α < 1/2, while such an assumption with α = 1/2 is not suffi- cient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The whole situation becomes very different when the cosine-function is replaced by a more general 1-periodic function, even if it is such a well-behaved one as a LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 13 trigonometric polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, consider (12) f(x) = cos(2πx) − cos(4πx), nk = 2k, k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In this case the lacunary sum is telescoping, and it can be immediately seen that there cannot be a non-trivial limit distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A more delicate example is attrib- uted to Erd˝os and Fortet2, and goes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let f(x) = cos(2πx) + cos(4πx), nk = 2k − 1, k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then it can be shown that N−1/2 �N k=1 f(nkx) does indeed have a limit distribution, but one which is actually non-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' More precisely, for this example one has lim N→∞ λ � x ∈ (0, 1) : N � k=1 f(nkx) ≤ t � N/2 � = 1 √π � 1 0 � t/2| cos(πs)| −∞ e−u2duds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus the limit distribution in this case is a so-called “variance mixture Gaussian”, which can be seen as a normal distribution whose variance is a function rather than a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This limiting behavior can be explained from the observation that (13) f(nkx) = cos((2k+1 − 2)πx) + cos((2k+2 − 4)πx) and (14) f(nk+1x) = cos((2k+2 − 2)πx) + cos((2k+3 − 4)πx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Combining the second term on the right-hand side of (13) with the first term on the right-hand side of (14) we obtain cos((2k+2 − 4)πx) + cos((2k+2 − 2)πx) = 2 cos(πx) cos((2k+2 − 3)πx), so the whole lacunary sum �N k=1 f(nkx) can essentially be written as 2 cos(πx) multiplied with a pure cosine lacunary sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This is exactly what the “variance mixture Gaussian” indicates: the limit distribution is actually that of 2 cos(πx) independently multiplied with a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The failure a of Gaussian central limit theorem in the example above can be seen as a consequence of the fact that the Diophantine equation nk+1 − 2nk = 1 possesses many solutions k for this particular choice of sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Equipped with this observation, one could readily construct similar examples with other trigono- metric polynomials f, and other variance mixture Gaussians as limit distributions, by creating situations where there are many solutions k, ℓ to (15) ank − bnℓ = c 2The Erd˝os–Fortet example is first mentioned in print in a paper of Salem and Zygmund [202].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' They mention the example without proof, and write: “This remark is essentially due to Erd˝os.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Later the example was mentioned in a paper of Kac [146], who wrote: “It thus came as a surprise when simultaneously and independently of each other, Erd˝os and Fortet constructed an example showing that the limit [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' ] need not be Gaussian”, with a footnote: “In Salem and Zygmund this example is erroneously credited to Erd˝os alone.” No proof is given in Kac’s paper either, but he writes: “Details will be given in [a forthcoming] paper by Erd˝os, Ferrand, Fortet and Kac”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Such a joint paper never appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 14 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY for some fixed a, b, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, interestingly, a special role is played by such equations when c has the particular value c = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' very roughly speaking, solutions of the equation for c = 0 effect only the limiting variance (in a Gaussian distribution), but not the structure of the limiting distribution itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This is visible in a paper of Kac [145], who studied the sequence nk = 2k, k ≥ 1, where indeed the only equations that have many solution are of the form 2mnk − nℓ = 0 for some m (the solutions being ℓ = k + m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Kac proved that for this sequence and any 1-periodic f of bounded variation and zero mean one has (16) lim N→∞ λ � x ∈ (0, 1) : N � k=1 f(nkx) ≤ tσf √ N � = Φ(t) with a limiting variance σ2 f, provided that (17) σ2 f := � 1 0 f 2(x) dx + 2 ∞ � m=1 � 1 0 f(x)f(2mx) dx ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus in this case the limit distribution is always a Gaussian, and the failure of the trivial example in (12) to produce such a Gaussian limit comes from the fact that the limiting variance is degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' These observations show that there is a delicate interplay between arithmetic, an- alytic and probabilistic effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' in particular, it is obviously not only the order of growth of (nk)k≥1 which is responsible for the fine probabilistic behavior of a la- cunary sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Takahashi [211] proved a CLT (with pure Gaussian limit) under the assumption that nk+1/nk → ∞, and Gaposhkin [128] proved that a CLT (with pure Gaussian limit) holds when nk+1/nk is an integer for all k, or if nk+1/nk → α for some α such that αr ̸∈ Q, r = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (and if additionally the variance does not degenerate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A general framework connecting Diophantine equations and the dis- tribution of lacunary sums was established in Gaposhkin’s profound paper [131], where he proved that a CLT (with pure Gaussian limit) holds if for all fixed positive integers a, b the number of solutions k, ℓ of the Diophantine equation (15) is bounded by a constant which is independent of c (where only c ̸= 0 needs to be considered, provided that the variance does not degenerate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' One can check the validity of this general condition for sequences satisfying the assumptions mentioned earlier in this paragraph, such as nk+1/nk → ∞ or nk+1/nk → α for αr ̸∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Finally, an optimal result was established in [13]: For (nk)k≥1 satisfying the Hadamard gap condition, the limit distribution of N−1/2 �N k=1 f(nkx) is Gaussian provided that the number of solutions (k, ℓ) of (15), subject to k, ℓ ≤ N, is of order o(N) (for all fixed a, b, uniformly in c ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If, on the other hand, for some a, b, c the number of solutions is Ω(N), then the CLT generally fails to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If the number of solutions with c = 0 also is of order o(N), then the CLT has the “correct” variance � 1 0 f(x)2dx, in perfect accordance with the independent case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Even if the number of solutions is of order Ω(N) for some a, b, c, then the deviation of the distribution of N−1/2 �N k=1 f(nkx) from the Gaussian distribution can be quantified in terms of the ratio “(number of solutions)/N”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This shows for example that while the CLT generally fails in the LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 15 case nk+1/nk → p/q, one obtains an “almost CLT” if both p and q are assumed to be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Another example for such an “almost CLT” is when the growth constant in the Hadamard gap condition is assumed to be very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See the statement of [13, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='3] and the subsequent discussion for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that Gaposhkin’s condition implies that the CLT also holds for all subsequences that are picked out of (nk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This is not the case under the assumptions from [13], where one might be able to extract a subsequence along which the CLT fails (by choosing a subsequence which allows a large number of solutions of the relevant Diophantine equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is interesting that the probabilistic behavior of lacunary sums might change when one passes to a subsequence of the original sequence – this is in clear contrast to the bevahior of sums of independent random variables, where any subsequence of course is independent as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A similar remark holds for permutations of lacunary sums resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' permutations of sums of independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' These phenomena have received strong attention during the last years;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see for example [17, 18, 19, 21, 114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' To give only one sample result, in [17] the following is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As noted above, the CLT is true for pure trigonometric sums under the Erd˝os gap condition nk+1/nk ≥ 1 + ck−α for some α < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, this is only true for the unpermuted sequence (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' sorted in increasing order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If permutations of the sequence are allowed, then this gap condition is not sufficient anymore for the validity of the CLT, as is no other gap conditon weaker than Hadamard’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' More precisely, for any sequence (εk)k≥1 with εk → 0 there exists a sequence of positive integers satisfying nk+1/nk ≥ 1 + εk, together with a permutation σ : N �→ N, such that the permuted (pure trigonometric) sum N−1/2 �N k=1 cos(2πnσ(k)x) converges in distribution to a non-Gaussian limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' One can also construct such examples where the norming sequence N−1/2 has to be replaced by (log N)1/2N−1/2 and the limit is a Cauchy distribution, and examples where no limit distribution exists at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See [17] for details on this particular result, and Chapter 3 of [61] for a detailed discussion of permutation-invariance of limit theorems for lacunary (trigonometric) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We close this section with some further references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For Hadamard lacunary (nk)k≥1, the limit distribution of N1/2DN(nkx) was calculated in [14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' under suitable Dio- phantine assumptions it coincides with the Kolmogorov distribution, which is the distribution of the range of a Brownian bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A central limit theorem for Hardy– Littlewood–P´olya sequences was established in [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In [87] the Erd˝os–Fortet ex- ample was revisited from the perspective of ergodic theory, and was interpreted in terms of the limiting behavior of certain modified ergodic sums, and generalized to cases such as expanding maps, group actions, and chaotic dynamical systems under the assumption of multiple decorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See also [86, 88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The limit distribution of N−1/2 �N k=1 cos(2πnkx) for the special sequence nk = k2, k ≥ 1, was determined by Jurkat and Van Horne in [141, 142, 143], and turned out to have finite moments of order < 4, but not of order 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The theory of such sums is closely related to theta sums, and goes back to Hardy and Littlewood [135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For further related results, see [80, 108, 219].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For non-Gaussian limit distributions of N−1/2 �N k=1 cos(2πnkx) 16 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY near the Erd˝os gap condition nk+1/nk ≥ 1 + ck−1/2 see [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For a multidimensional generalization of Kac’s results see [112, 120], and for a multidimensional generaliza- tion of the CLT for Hardy–Littlewood–P´olya sequences (considering a semi-group generated by powers of matrices instead) see [167, 168].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See also [85, 90] for general- izations of the CLT for Hardy–Littlewood–P´olya sequences to a very general setup of sums over powers of transformations/automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The law of the iterated logarithm for lacunary sequences Together with the law of large numbers (LLN) and the central limit theorem (CLT), the law of the iterated logarithm (LIL) is one of the fundamental results of prob- ability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Very roughly speaking, the (strong) law of large numbers says that when scaling by N−1 one has almost sure convergence of a sum of random variables, and the central limit theorem says that when scaling by N−1/2 one has a (Gaussian) limit distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The law of the iterated logarithm operates between these two other asymptotic limit theorems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' in its simplest form, it says that for a sequence (Xn)n≥1 of centered i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables (under suitable extra assumptions, such as boundedness) one has lim sup N→∞ �N n=1 Xn √2N log log N = σ almost surely, where σ is the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Heuristically, the law of the iterated logarithm identifies the threshold between convergence in distribution and almost sure conver- gence for sums of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' indeed, while �N n=1 Xn √2N log log N converges to 0 in distribution by the CLT, it does not converge to 0 almost surely by the LIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The first version of the LIL was given by Khinchin in 1924, and a more general variant was established by Kolmogorov in 1929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that the law of large numbers for trigonometric sums or sums of dilated functions is rather unproblematic: for any sequence of distinct integers (nk)k≥1 one has (18) lim N→∞ 1 N N � k=1 f(nkx) = � 1 0 f(x) dx, as long as one can assume a bit of regularity for f (such as f being a trigonometric polynomial, being Lipschitz-continuous, being of bounded variation on [0, 1], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Only if one is not willing to impose any regularity assumptions upon f the situation becomes quite different;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see the remarks at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The most basic law of the iterated logarithm for lacunary systems is (19) lim sup N→∞ �N n=1 cos(2πnkx) √2N log log N = 1 √ 2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' under the Hadamard gap condition on (nk)k≥1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' this was obtained by Salem and Zygmund (upper bound) [203] and Erd˝os and G´al (lower bound) [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Generally speaking, as often in probability theory the lower bound is more difficult to estab- lish than the upper bound, since the latter can be proved by an application of the LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 17 first Borel–Cantelli lemma (convergence part), while the former is proved by the second Borel–Cantelli lemma (divergence part, which needs some sort of stochastic independence as an extra assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that (19) is a perfect analogue of (18) with the “correct” constant on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As in the case of the CLT, replacing pure trigonometric sums by sums of more general 1-periodic functions makes the situation much more delicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As in the previous section, a key role is played by Diophantine equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, while for the CLT it is crucial that the number of solutions of Diophantine equations “stabilizes” in some way to allow for a limit distribution (albeit a potentially non- Gaussian one), no such property is necessary for the validity of a form of the LIL (since, as noted above, this is defined as a lim sup, not as a lim).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Instructive examples are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In all examples, we assume that f is 1-periodic with mean zero and bounded variation on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If nk+1/nk → ∞ as k → ∞, then lim sup N→∞ �N n=1 f(nkx) √2N log log N = �� 1 0 f 2(x) dx �1/2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If nk = 2k, k ≥ 1, then lim sup N→∞ �N n=1 f(nkx) √2N log log N = σf a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', with σ2 f = � 1 0 f 2(x)dx + 2 ∞ � m=1 � 1 0 f(x)f(2mx) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Assume that nk+1/nk ≥ q > 1, k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then there exists a constant C (depending on f and on q) such that (20) lim sup N→∞ �N n=1 f(nkx) √2N log log N ≤ C a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If nk = 2k − 1, k ≥ 1, and if f(x) = cos(2πx) + cos(4πx), then (21) lim sup N→∞ �N n=1 f(nkx) √2N log log N = √ 2| cos(πx)| a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The first result in this list (due to Takahashi [213]) is in perfect accordance with the LIL for truly independent random sums, in accordance with the fact that also the CLT holds in the “truly independent” form under the large gap condi- tion nk+1/nk → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The second result is an analogue of Kac’s CLT in Equations (16) and (17): as with the CLT, also the LIL holds for the sequence (2k)k≥1, but the limiting variance deviates from the one in the “truly independent” case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that in contrast to the CLT case we now do not need to require that σf ̸= 0 for the validity of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The third result (Takahashi [212]) asserts that there is an upper-bound version of the LIL for lacunary sums (even for sequences where there is no convergence of distributions, and any form of the CLT fails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Finally, the fourth 18 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY result (the Erd˝os–Fortet example for the LIL instead of the CLT) shows the remark- able fact that the lim sup in the LIL for Hadamard lacunary sums might actually be non-constant – this is very remarkable, and a drastic deviation from what one can typically observe for sequences of independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In particular this example shows that under the Hadamard gap condition an upper-bound version of the LIL is in general the best that one can hope for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Not very surprisingly, the source of all these phenomena are (as in the previous section) Diophantine equations such as (15), and their number of solutions within the sequence (nk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' So in the LIL there is again a complex interplay between probabilistic, analytic and arithmetic aspects which controls the fine asymptotic behavior of lacunary sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In probability theory there is a version of the LIL for the Kolmogorov–Smirnov statistic of an empirical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This is called the Chung–Smirnov LIL, and in the special case of a sequence (Xn)n≥1 of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables having uniform distribution on [0, 1] (where the Kolmogorov–Smirnov statistic coincides with the discrepancy) it asserts that lim sup N→∞ NDN(Xn) √2N log log N = 1 2 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here the number 1/2 on the right-hand side arises essentially as the maximal L2 norm (“standard deviation”) of a centered indicator function of an interval A ⊂ [0, 1] (namely the indicator function of an interval of length 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Based on the princi- ple that lacunary sequences tend to “imitate” the behavior of truly independent sequences, it was conjectured that an analogue of the Chung–Smirnov LIL should also hold for the discrepancy of ({nkx})k≥1, where (nk)k≥1 is a Hadamard lacunary sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This was known as the Erd˝os–G´al conjecture, and was finally solved by Philipp [193], who proved that for any q > 1 there exists a constant Cq such that for (nk)k≥1 satisfying nk+1/nk ≥ q we have (22) 1 √ 32 ≤ lim sup N→∞ NDN({nkx}) √2N log log N ≤ Cq a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' An admissible value of Cq was specified in [193] as Cq = 166/ √ 2 + 664/(√2q − √ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The first inequality in (22) follows from (a complex version of) Koksma’s inequality together with (19), so the novelty is the second inequality (upper bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note also that the upper bound in (22) implies Takahashi’s “upper bound” LIL in (20), again as a consequence of Koksma’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Philipp’s result has been extended and refined into many different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The most precise results, many of which were obtain by Fukuyama, show again a fasci- nating interplay between arithmetic, analytic and probabilistic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As a sample we state the following results (all from Fukuyama’s paper [113]): Let nk = θk, k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then (23) lim sup N→∞ NDN({nkx}) √2N log log N LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 19 exists and is constant (almost everywhere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Denoting the value of this lim sup by Σθ, we have: If θr ̸∈ Q for all r = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , then Σθ = 1/2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If r denotes the smallest positive integer such that θr = p/q for some coprime p, q, then 1/2 ≤ Σθ ≤ � (pq + 1)/(pq − 1)/2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If θr = p/q as above and both p and q are odd, then Σθ = � (pq + 1)/(pq − 1)/2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If θ = 2, then Σθ = √ 42/9 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If θ > 2 is an even integer, then Σθ = � (p + 1)p(p − 2)/(p − 1)3/2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If θ = 5/2, then Σθ = √ 22/9 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' All these results were obtained by very delicate calculations involving Fourier anal- ysis and Diophantine equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The calculations from [113] were continued by the same author and his group in [119, 121, 122, 125], so that now we have a relatively comprehensive picture on the behavior of these lim sup’s in the case when (nk)k≥1 is (exactly) a geometric progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In [7, 8] for general Hadamard lacunary sequences (nk)k≥1 a direct connection was established which links the number of solutions of (15) with the value of the lim sup in the LIL, in the same spirit as this was done before in [13] for the CLT (as de- scribed in the previous section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In particular, if the number of solutions of (15) is sufficiently small, then the LIL holds with the constant 1/2 on the right-hand side, exactly as in the truly independent case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Another interesting observation is that if (nk)k≥1 is Hadamard lacunary with growth factor q > 1, and if Σ denotes the value of the lim sup in (23), then the difference |Σ − 1/2| can be quantified in terms of q and tends to zero a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' as q → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus there is a smooth transition towards the “truly independent” LIL as the growth factor q increases, and under the large gap condition nk+1/nk → ∞ the value of Σ actually equals 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Another remark- able fact is that there exist Hadamard lacunary sequences for which the lim sup in the LIL for the discrepancy is not a constant almost everywhere, but rather a function of x, similar to what happened in (21) for the LIL for � f(nkx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In some cases the limit functions in the LIL for the discrepancy can be explicitly calculated, and are “surprisingly exotic” (in the words of Ben Green’s MathSciNet review of [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As noted above, Philipp’s LIL for the discrepancy has been extended into many different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, while it is known that the result can fail as soon as the Hadamard gap condition is relaxed to any sub-exponential growth condition, it turns out to be possible to obtain an LIL for the discrepancy when a weaker growth condition is compensated by stronger arithmetic assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In particu- lar, an analogue of Philipp’s result has been proved for Hardy–Littlewood–P´olya sequences [195];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see also [5, 65, 123, 215].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As a closing remark concerning the LIL, it is interesting that the optimal value of the lower bound in (22) is still unknown;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [25] for more context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 20 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY While much effort has been spent towards understanding the probabilistic behavior of lacunary sums at the scales of the CLT and LIL, it seems that investigations at other scales (such as in particular at the large deviations scale) were only started recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The few results which are currently available point once again towards an intricate connection between probabilistic, analytic and arithmetic effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see the very recent papers [22, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Normality and pseudorandomness Normal numbers were introduced by Borel [73] in 1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' From the very beginning the concept of normality of real numbers was associated with “randomness”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' While normality of real numbers was originally defined in terms of counting the number of blocks of digits, it is not difficult to see3 that a number x is normal in base b if and only if the sequence ({bnx})n≥1 is equidistributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As Borel proved, Lebesgue- almost all real numbers are normal in a (fixed) integer base b ≥ 2, and thus almost all reals are normal in all bases b ≥ 2 (such numbers are called absolutely normal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' While normal numbers are ubiquitous from a measure-theoretic perspective,4 it is difficult to construct normal numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The most fundamental construction is due to Champernowne, who proved (using combinatorial arguments) that the number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='1 2 3 4 5 6 7 8 9 10 11 12 13 14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , which is obtained by a concatenation of the decimal expansions of the integers, is normal in base 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The idea of creating normal numbers by a concatenation of the (b-ary) expansions of the values of (simple) functions at integers (or primes) is still the most popular, and probably most powerful, method in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We only note that Copeland and Erd˝os [89] proved that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='2 3 5 7 11 13 17 19 23 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , which is obtained by concatenating the decimal expansions of the primes, is normal in base 10, and refer to [92, 93, 171, 174, 186, 187] for more results of this flavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It should be noted, however, that there have been earlier constructions of a con- ceptually very different nature, such as that of Sierpinski [208] in 1917.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See [43] for an exposition of Sierpinski’s construction and more context, and see also [44] on an early (unpublished) algorithm of Turing for the construction of normal numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We finally mention a very recent idea for the construction of a normal number by Drmota, Mauduit and Rivat [95], which is not based on the concatenation of deci- mal blocks as above, but rather on the evaluation of an automatic sequence along a subsequence of the index set (in this particular case, the Thue–Morse sequences evaluated along the squares);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see also [183, 210].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 3Probably first explicitly mentioned by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Wall in his PhD thesis, 1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 4Interestingly, while the set of normal numbers is large from a measure-theoretic point of view, it turns out to be small from a topological point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' More precisely, the set of normal numbers is meager (of first Baire category), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In the words of Edmund Hlawka [139, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 78]: “Thus whereas the normal numbers almost force themselves on to the measure theorist, the topologist is apt to overlook them entirely.” LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 21 While Lebesgue-almost all numbers are normal and there are some constructions of normal numbers, it is generally considered to be completely hopeless to prove that natural constants such as π, e, √ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' are normal in a given base (although the experimental evidence clearly points in that direction: [189, 218, 224]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Many such open problems “for the next millennium” are contained in Harman’s survey article [137];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see also [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, it is quite clear that the mathematical machin- ery which would be necessary to prove the normality of √ 2 or other such constants is completely lacking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' compare the rather deplorable current state of knowledge on the binary digits of √ 2 as given in [34, 98, 217].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A small spark of hope is provided by the very remarkable formulas of Bailey, Borwein and Plouffe (now widely known as BBP formulas), which allow to calculate deep digits of π (and other constants) without the need of computing all previous digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See [51] for a very comprehen- sive “source book” covering computational aspects of π, and see [35] for a very rare example of a possible strategy of what a proof of the normality of π could possibly like (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' also [162]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Since normality of x in a base b can be expressed in terms of the equidistribution of the sequence ({bnx})n≥1, it is very natural to consider the discrepancy of DN({bnx}) and call this (with a slight abuse of language) the discrepancy of x (as a normal number, with respect to a base b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Remarkably, it is still unknown how small the discrepancy of a normal number can be (this is known as Korobov’s problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Levin [166] constructed (for given base b) a number x such that DN({bnx}) = O �(log N)2 N � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' by Schmidt’s general lower bound the exponent of the logarithm cannot be reduced below 1, but the optimal size of this exponent remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' One of the most interesting, and most difficult, aspects of normal numbers is nor- mality with respect to two or more different bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Extending work of Cassels [79], Schmidt [206] characterized when normality with respect to a certain base im- plies normality with respect to another base, and when this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See also [48, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, generally speaking it is very difficult to construct numbers which are normal with respect to several different bases, and the “constructions” are much less explicit than the ones of Champernowne and Copeland–Erd˝os men- tioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The problem of the minimal order of the discrepancy of normal num- bers seems to be very difficult when different bases are considered simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Aistleitner, Becher, Scheerer and Slaman [12] constructed a number x such that DN({bnx}) = Ob � N−1/2� for all integer bases b ≥ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' this is considered to be an “unexpectedly small” order of the discrepancy by Bugeaud in his MathSciNet review of [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is not known if the exponent −1/2 of N in this estimate is optimal or not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' indeed, no non-trivial lower bounds whatsoever (beyond the general lower bound (log N)/N of Schmidt) 22 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY are known for this problem, but it is quite possible that whenever simultaneous nor- mality with respect to different (multiplicatively independent) bases is considered, there must be at least one base for which the discrepancy is “large”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In a recent years, there has been a special focus on algorithmic aspects of the con- struction of normal numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A particularly striking contribution was a polynomial- time algorithm for the construction of absolutely normal numbers due to Becher, Heiber and Slaman [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See also [29, 47, 205].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Related to such algorithmic and computational problems are questions on the complexity of the set of normal num- bers from the viewpoint of descriptive set theory in mathematical logic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' in this framework, the rank of the set of normal numbers [152] and absolutely normal num- bers [46] within the Borel hierarchy has been determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The notion of normality can be extended in a natural way to many other situa- tions, where it is always understood that normality should be the typical behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, one can consider normal continued fractions, where the “expected” number of occurences of each partial quotient is prescribed by the Gauss–Kuzmin measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see for example [1, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Other generalizations consider for example normal- ity with respect to β-expansions [39, 173], a numeration system which generalizes the b-ary expansion to non-integral bases β, or normality with respect to Cantor ex- pansions [3, 109, 175], a numeration system which allows a different set of “digits” at each position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For a particularly general framework, see [172].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Interestingly, in such generalized numeration systems there can be more than one natural definition of normality, using as starting point for exampe either the idea of counting blocks of digits, or the idea of equidistribution of an associated system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The relation be- tween such different (sometimes conflicting) notions of normality has been studied in particular detail for Cantor expansions [2, 170, 176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Normal numbers feature prominently in the chapter on random numbers in Volume 2 of Knuth’s celebrated series on The Art of Computer Programming [153].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' There he tries to come to terms with the notion of “random” sequences of numbers, and intro- duces an increasingly restrictive scheme of “randomness” of deterministic sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The concept of normality is also the starting point for one of the (quantitative) mea- sures of pseudorandomness, which were introduced by Mauduit and S´ark¨ozy [179] and then studied in a series of papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note in this context that the transformation T : x �→ bx mod 1, which is at the foundation of the concept of normal numbers, can in some sense be seen as the continuous analogue of the recursive formula which defines a linear congruential generator (LCG), one of the most classical devices for pseudorandom number generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For this connection between normal numbers and pseudo-random number generators, see for example [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Another very fruitful aspect of normal numbers is the connection with ergodic theory, which comes from the observation that the sequence ({bnx})n≥1 is the orbit of x under the transforma- tion T from above, and that this transformation is measure-preserving (with respect to the Lebesgue measure) and ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We will not touch upon this connection in LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 23 any detail, and instead refer to [91, 149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Another sequence which is often associated with “randomness” is the sequence ({xn})n≥1 for real x > 1, or more generally ({ξxn})n≥1 for ξ ̸= 0 and x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This looks quite similar to a (Hadamard) lacunary sequence such as (bnx)n≥1, but its metric theory is of a very different nature in several respects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Both sequences are variants of a geometric progression, but while in the lacunary sequence the base b is fixed and x is assumed to be a “parameter”, now ξ is assumed to be fixed and the base x of the geometric progression is the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' While (bnx)n≥1 can in many ways be easily interpreted in terms of harmonic analysis, digital expansions, ergodic theory, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', such simple interpretations fail for ({xn})n≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note in particular that in contrast to lacunary sequences there now is no periodicity when replacing x �→ x+1, there is no “orthogonality”, and the calculation of moments of sums � f(xn) does not simply reduce to the counting of solutions of Diophantine equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Still, what is preserved from the setup of lacunary sequences is that xn (as a function of x) oscillated quickly on intervals where xm is essentially constant, provided that n is significantly larger than m, and there are good reasons to consider systems such as (cos(2πxn))n≥1 to be “quasi-orthogonal” and “almost independent” in some appro- priate sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' One of the most fundamental results on this type of sequence is due to Koksma [154]: assuming that ξ ̸= 0 is fixed, the sequence ({ξxn})n≥1 is uniformly distributed mod 1 for almost all x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In particular, when ξ = 1, the sequence ({xn})n≥1 is uniformly distributed mod 1 for almost all x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In very sharp contrast with Koksma’s metric result is the fact that until today not a single example of a number x is known for which ({xn})n≥1 is indeed uniformly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This problem is related with Mahler’s problem on the range of ({(3/2)n})n≥1, which also seems to be completely out of reach for current methods (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [97, 110]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The sequence ({xn})n≥1 is discussed at length in Knuth’s book, where it is conjectured that this sequence is a good candidate to pass several very strict pseudorandomness criteria for almost all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, Knuth conjectured that for all sequences of distinct integers (sn)n≥1 the sequence ({xsn})n≥1 (a subsequence of the original sequence) has a strong equidistribution property called complete uniform distribution, for almost all x > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' this was indeed established by Niederreiter and Tichy [188].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is also known that ({xn})n≥1 satisfies an law of the iterated logarithm in the “truly independent” form lim sup N→∞ NDN({xn}) √2N log log N = 1 2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', and similarly satisfies a central limit theorem which is perfectly analogous to the one for truly independent systems [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that Knuth’s assertion that the sequence ({xn})n≥1 shows good pseudo-random behavior for almost all x > 1 is of limited practical use, as long as no such value of x is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The discrete analogue would be to study the pseudo-randomness properties of an mod q for n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , where a and q are fixed integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Investigations on the pseudo-randomness properties of such sequence were carried out for example by Arnol’d [32], who experimentally observed 24 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY good pseudorandom behavior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' also [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' To close this section, we note that equidistribution is of course just one property which can be used to characterize “pseudorandom” behavior (essentially by analogy with the Glivenko–Cantelli theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' There are many other statistics which could be applied to a sequence in [0, 1] to determine whether it behaves in a “random” way or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' One class of such statistics are gap statistics at the level of the average gap (which is of order 1/N when considering the first N elements of a sequence in [0, 1]), such as the distribution of nearest-neighbor gaps, or the pair correlation statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We do not give formal definitions of these concepts here, but note that they are inspired by investigations of the statistics of quantum energy eigenvalues in the context of the Berry–Tabor conjecture in theoretical physics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see [177] for more context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Pseudorandom behavior with respect to such statistics is called “Poisso- nian”, since it agress with the corresponding statistics for the Poisson process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The general principle that lacunary systems show pseudorandom behavior is also valid in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, Rudnick and Zaharescu [199] showed that for (nk)k≥1 satisfying the Hadamard gap condition the sequence ({nkx})k≥1 is Poissonian for almost all x, and Aistleitner, Baker, Technau and Yesha [11] showed that the same holds for ({xn})n≥1 for almost all x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This section on normal numbers and sequences of the form ({ξxn})n≥1 gives of course only a very brief overview of the subject, and has to leave out many interesting aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For a much more detailed exposition we refer the reader to the book of Bugeaud [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Random sequences In the previous sections we have illustrated the philosophy that gap sequences ex- hibit many probabilistic properties which are typical for sequences of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In many cases the large gap condition nk+1/nk → ∞ gives “true” random limit theorems, the Hadamard gap condition nk+1/nk ≥ q > 1 is a critical transi- tion point where a mixture of probabilistic, analytic and arithmetic effects comes into play, and the “almost independent” behavior is lost when the gap condition is relaxed below Hadamard’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' There are results which hold under weaker gap condi- tions such as the Erd˝os gap condition nk+1/nk ≥ 1 + ck−α, 0 < α < 1/2, or under additional arithmetic assumptions, but as a whole the Hadamard gap condition is the critical point where the “almost independent” behavior of systems of dilated functions starts to break down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, while almost independent behavior is generally lost under a weaker gap condition (without strong arithmetic information), there is another possible per- spective on the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As noted, for a fixed sequence (nk)k≥1 one cannot ex- pect “almost independent” behavior of ({nkx})k≥1, say, without assuming a strong growth condition on (nk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, even without such a growth condition one might expect that ({nkx})k≥1 shows independent behavior for “typical” sequences (nk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here the word “typical” of course implies that the sequence has to be taken LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 25 from a generic set in some appropriate space which possesses a measure, so quite naturally this idea leads to considering “random” sequences (nk)k≥1 = (nk(ω))k≥1 which are constructed in a randomized way over some probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Of course there are many possible ways how a random sequence can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' From results of Salem and Zygmund [204] for trigonometric sums with random signs it follows easily that if we define a sequence (nk)k≥1 by flipping a coin (independently) for every positive integer to decide whether it should be contained in the sequence or not,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' and let P denote the probability measure on the space over which the “coins” are defined,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' then for P-almost all sequences as defined above one has (24) 1 √ N N � k=1 cos(2πnkx) D −→ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 1/4) and (25) lim sup N→∞ 1 √2N log log N N � k=1 cos(2πnkx) = 1 2 for almost all x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' where N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' σ2) denotes the normal distribution with mean 0 and variance σ2 and D −→ denotes convergence in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that (24) and (25) are not exactly matching with the truly independent case, where the limit distribution would be N(0, 1/2) and the limsup in the LIL would be 1/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The “loss” on the right-hand sides of (24) and (25) comes from the fact that a Dirichlet kernel is “hiding” in this linearly growing sequence, and this kernel is highly localized near 0 and 1 so that its contribution is lost in the CLT and LIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By the strong law of large numbers (SLLN) clearly nk ∼ 2k as k → ∞, P-almost surely, so the sequences constructed here are very far from satisfying any substantial gap condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' in contrast, their (typical) order of growth is only linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It should be noted that the gaps nk+1 − nk in this sequence are not bounded: with full P-probability, nk+1 −nk = 1 for infinitely many k (roughly, in half of the cases), but for infinitely many k, the gap nk+1 − nk has order of magnitude c log k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' this follows from the “pure heads” theorem of Erd˝os and R´enyi, see [198].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We call an increasing sequence (nk)k≥1 of positive integers a B2 sequence if there exists a constant C > 0 such that for any integer ν > 0 the number of representations of ν in the form ν = nk ±nℓ, k > ℓ ≥ 1, is at most C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By a result of Gaposhkin [131] already mentioned in Section 4, the sequence (f(nkx))k≥1 satisfies the CLT for all Hadamard lacunary (nk)k≥1 and all 1-periodic Lipschitz continuous f if and only if for any m ≥ 1, the set-theoretic union of the sequences (nk)≥1, (2nk)≥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , (mnk)≥1 satisfies the B2 condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='5 No similarly complete result is known for sequences (nk)k≥1 growing slower than exponentially, but Berkes [54] proved that if (nk)k≥1 is 5Note that the definition of the B2 property used in [131] is slightly different from the standard usage in number theory (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [134]) requiring that the number of solutions of ν = nk + nℓ, k > ℓ ≥ 1, is bounded by C, but this does not affect the discussion below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 26 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY a B2 sequence satisfying the gap condition (26) nk+1/nk ≥ 1 + ck−α, k ≥ 1, for some c > 0, α > 0, then (cos(2πnkx))k≥1 satisfies the CLT and LIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' To verify the B2 property for a concrete sequence (nk)k≥1 is generally a difficult problem, but the situation is quite different for random constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let I1, I2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' be disjoint blocks of consecutive integers and let n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' be independent random variables on some probability space (Ω, F, P) such that nk is uniformly distributed over Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Clearly, the number of different sums ±nk1 ± nk2 ± nk3, 1 ≤ k1, k2, k3 ≤ k − 1, is at most 8(k − 1)3, and thus if the size of Ik is ≥ k5, then the probability that nk is equal to any of these sums is ≤ 8k3k−5 = O(k−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus by the Borel-Cantelli lemma, with P-probability 1, such a coincidence can occur only for finitely many k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus the equation ±nk1 ± nk2 ± nk3 ± nk4 = 0, k1 ≤ k2 ≤ k3 < k4 has only finitely many solutions, which implies that (nk)k≥1 is a B2 sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let us recall now that by a result of Erd˝os [102], (cos(2πnkx))k≥1 satisfies the CLT with limit distribution N(0, 1/2), provided that (26) holds with α < 1/2, and this result is sharp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' there exists a sequence (nk)k≥1 satisfying (26) with α = 1/2 such that the CLT fails for (cos(2πnkx))k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that the counterexample is irregular: while nk+1/nk − 1 is of the order O(k−1/2) for most k, there is also a subsequence along which nk+1/nk → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' One may therefore wonder if regular behavior of nk+1/nk implies the CLT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' in particular, Erd˝os [102] conjectured that the CLT holds for (cos(2πnkx))k≥1 if nk = ⌊e(kβ)⌋ for some β in the range 0 < β ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (Note that for β > 1/2 condition (26) is satisfied with α < 1/2, so the CLT follows from Erd˝os’ result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') This conjecture was proved by Murai [184] for β > 4/9, but for smaller β the problem is still open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Random constructions provide here important information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Kaufman [148] proved that if c is chosen at random, with uniform distribution on a finite interval (a, b) ⊂ (0, ∞), then (cos(2πnkx))k≥1 with nk = e(ckβ) satisfies the CLT with probability 1 for any fixed β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' An even wider class of random sequences with the CLT property is obtained by choosing the blocks Ik in the random construction above as the integers in the interval (27) Jk = � e(ckβ)(1 − rk), e(ckβ)(1 + rk) � , rk = o(k−(1−β)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A simple calculation shows that these intervals are disjoint for k ≥ k0 and for nk ∈ Jk we have (26) with α = 1 − β, in fact we even have nk+1/nk = 1 + c1(1 + o(1))/k1−β with some constant c1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Now if rk decreases like a negative power of k, then the length of Jk will be ≥ k5 and thus the constructed random sequence (nk)k≥1 will be a B2 sequence with probability 1, so (cos(2πnkx))k≥1 satisfies the CLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In other words, the CLT for (cos(2πnkx))k≥1 holds for a huge class of sequences nk ∼ e(ckβ) for any c > 0, β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 27 Concerning B2 sequences, it is worth pointing out that Erd˝os [101] proved, decades before Carleson’s convergence theorem, that �∞ k=1(ak cos(2πnkx) + bk sin(2πnkx)) is almost everywhere convergent if (nk)k≥1 is a B2 sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The question of how slowly B2 sequences can grow is a much investigated problem of number theory, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Halberstam and Roth [134], Chapters II and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is easily seen that a B2 sequence (nk)k≥1 cannot be o(k2) and Erd˝os and R´enyi [105] proved by a random construction that for any ε > 0 there exists a B2 sequence (nk)k≥1 with nk = O(k2+ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Changing the B2 property slightly and requiring that all numbers nk ± nℓ, k > ℓ, are actually different, makes the problem considerably harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The “greedy algorithm” yields a B2 sequence (nk)k≥1 with nk = O(k3), see [180], and it took nearly 40 years to improve this to nk = o(k3), see [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The best currently known (random) construction is due to Ruzsa [200], and satisfies nk = k1/( √ 2−1)+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (ωn)n≥1 be a nondecreasing sequence of positive integers tending to +∞ and let us divide the set of positive integers into disjoint blocks I1, I2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' such that the cardi- nality of Ik is ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Using these blocks in the random construction above, the resulting random sequence (nk)k≥1 cannot be a B2 sequence if (ωn)n≥1 grows slower than any power of n, but it is proved in Berkes [55] that with P-probability 1, (cos(2πnkx))k≥1 still satisfies the CLT and LIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The limit distribution here is N(0, 1/2) and the lim- sup in the LIL is 1/ √ 2, so that the “loss of mass” phenomenon observed in the case of the random sequence (nk)k≥1 in the Salem-Zygmund paper [204] does not occur here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The gaps in this sequence satisfy nk+1 − nk ≤ 2ωk+1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' they can grow arbitrarily slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' An LIL for the discrepancy of ({nkx})k≥1 under the same gap condition was given in Fukuyama [118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In [55] the question was raised if there exists a sequence (nk)k≥1 with bounded gaps nk+1 − nk = O(1) such that the CLT holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Bobkov and G¨otze [71] showed that if we want no loss of mass in the CLT, the answer is negative: if (nk)k≥1 is any increasing sequence of positive integers with nk+1 − nk ≤ L, k ≥ 1, such that N−1/2 �N k=1 cos(2πnkx) has a Gaussian limit distribution N(0, σ2), then necessarily σ2 < 1/2 and L ≥ 1/(1 − 2σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the other hand, Fukuyama [115, 116, 117] showed that for any σ2 < 1/2 there exists indeed a random subsequence (cos(2πnkx))k≥1 of the trigonometric system satisfying the CLT with limit distribution N(0, σ2) and with bounded gaps nk+1 − nk ≤ L with L ∼ 4/(1 − 2σ2) as σ2 → 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This shows that the result of Bobkov and G¨otze is optimal up to a factor 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This remarkable result is the “small gaps” counterpart of Erd˝os’ central limit theorem [102]: the latter determines the smallest gap sizes in (nk)k≥1 implying the CLT for (cos(2πnkx))k≥1, while Fukuyama’s result determines the smallest gap size which still allows a CLT with limit distribution N(0, σ2) to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is worth pointing out that the bounded gap sequences in [115, 116, 117] are obtained by rather complicated random constructions, while using the previously discussed simple construction and choosing the nk as independent random variables uniformly distributed over adjoining blocks Ik with equal length results in a random 28 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY sequence (nk)k≥1 satisfying almost surely (28) 1 √ N N � k=1 cos(2πnkx) D −→ N(0, Y ), where Y ≥ 0 is a random variable and N(0, Y ) is a “variance mixture” normal distribution with characteristic function E exp(−Y t2/2), see [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We also note that there is generally no “loss of mass” phenomenon for the LIL for trigonometric series with bounded gaps, see [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For further results for trigonometric series with bounded/random gaps, see [42, 41, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The subsequence principle The purpose of the previous sections was to illustrate the principle that thin subse- quences of the trigonometric system, or thin subsequences of a more general system of dilated functions, exhibit properties which are typical for sequences of indepen- dent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, an analogous principle holds in a much wider framework: it is known that, under suitable technical assumptions, sufficiently thin subsequences of general systems of random variables behave like genuine indepen- dent sequences, in the sense that a general sequence of random variables allows to extract a subsequence showing independent behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, Gaposhkin [128, 132] and Chatterji [83, 84] proved that if (Xn)n≥1 is any sequence of random variables satisfying supn EX2 n < ∞, then there exist a subsequence (Xnk)k≥1 and random variables X ∈ L2, Y ∈ L1, Y ≥ 0, such that (29) 1 √ N � k≤N (Xnk − X) D −→ N(0, Y ) and (30) lim sup N→∞ 1 √2N log log N � k≤N (Xnk − X) = Y 1/2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', where as at the end of the previous section N(0, Y ) denotes the “variance mix- ture”normal distribution with characteristic function E exp(−Y t2/2), and where again D −→ denotes convergence in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A functional (Strassen type) ver- sion of (30) was proved by Berkes [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By a result of Koml´os [159], from any sequence (Xn)n≥1 of random variables satisfying supn E|Xn| < ∞ one can select a subsequence (Xnk)k≥1 such that (31) lim N→∞ 1 N � k≤N Xnk = X a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for some X ∈ L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Chatterji [81] proved that if (Xn)n≥1 is a sequence of random vari- ables satisfying supn E|Xn|p < ∞ for some 0 < p < 2, then there exist a subsequence LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 29 (Xnk)k≥1 and a random variable X with E|X|p < ∞ such that (32) lim N→∞ 1 N1/p � k≤N (Xnk − X) = 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' These results establish the analogues of the central limit theorem (CLT), the law of the iterated logarithm (LIL), the strong law of large numbers (SLLN) and Mar- czinkiewicz’ strong law for subsequences (Xnk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note the mixed (or randomized) character of (29)–(32): the limit X in the strong law of large numbers, the cen- tering factor X in Marczinkiewicz’ strong law, and the limiting variance Y in the CLT (which also determines the limsup in the LIL) all become random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For fur- ther limit theorems for subsequences of arbitrary random variable sequences, see Gaposhkin [128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the basis of these and several other examples, Chatterji [82] formulated the following heuristic principle: Subsequence Principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let T be a probability limit theorem valid for all se- quences of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables belonging to an integrability class L defined by the finiteness of a norm ∥ ·∥L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then if (Xn)n≥1 is an arbitrary (dependent) sequence of random variables satisfying supn ∥Xn∥L < +∞ then there exists a subsequence (Xnk)k≥1 satisfying T in a mixed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In a profound study, Aldous [27] proved the validity of the subsequence principle for all distributional and almost sure limit theorems subject to minor technical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' To formulate his results, let M denote the class of probability measures on the Borel sets of R, equipped with the L´evy metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By [27], a subset A ⊂ M×R∞ is called a limit statute if: (a) P((λ, X1(ω), X2(ω), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') ∈ A) = 1 provided X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random vari- ables with distribution λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (b) (λ, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') ∈ A and � |xi − x′ i| < ∞ implies that (λ, x′ 1, x′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' An a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' limit theorem can thus be identified with a limit statute, where the analytic statement of the theorem is expressed by (a), while relation (b) means that a small perturbation of the sequence X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' does not change the validity of the limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let us give two examples of limit statutes representing the strong law of large numbers and the law of the iterated logarithm: A1 = � (λ, x) ∈ A : limN→∞ N−1 �N k=1 xk = |λ|1 � ∪ {(λ, x) : |λ|1 = ∞}, A2 = � (λ, x) ∈ A : lim supN→∞(2N log log N)−1/2 ��N k=1 xk − N|λ|1 � = |λ|2 � ∪ {(λ, x) : |λ|2 = ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here |λ|1 and |λ|2 denote the mean and variance of λ provided they are finite, and we write |λ|1 = ∞, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' |λ|2 = ∞ if � R |x|dλ(x) = ∞, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' � R |x|2dλ(x) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 30 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY On the other hand, by the definitions in [27], a weak limit theorem for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables is a system T = (f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , {Gλ, λ ∈ M0}) where (a) M0 is a measurable subset of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (b) For each k ≥ 1, fk = fk(λ, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') is a real function on M × R∞, measurable in the product topology, satisfying the smoothness condition |fk(λ, x) − fk(λ, x′)| ≤ ∞ � k=1 ck,i|xi − x′ i| where 0 ≤ ck,i ≤ 1 and limk→∞ ck,i = 0 for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (c) For each λ ∈ M0, Gλ is a probability distribution on the real line such that the map λ → Gλ is measurable (with respect to the Borel σ-field in M0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (d) If λ ∈ M0 and X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' are independent random variables with common distribution λ then fk(λ, X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , ) D −→ Gλ as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, the central limit theorem corresponds to the case when M0 is the class of distributions with mean 0 and finite variance, (33) fk(λ, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') = x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' + xk − kE(λ) √ k and Gλ = N(0, Var(λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let now (Xn)n≥1 be a sequence of random variables with supn ∥Xn∥L < ∞ with any norm ∥ · ∥L on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then (Xn)n≥1 is bounded in probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' lim K→∞ P(|Xn| > K) = 0 uniformly in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By an extension of the Helly–Bray theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [66]), (Xn)n≥1 has a subsequence (Xnk)k≥1 having a limit distribution conditionally on any event in the probability space with positive probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for any A ⊂ Ω with P(A) > 0 there exists a distribution function FA such that lim k→∞ P(Xnk ≤ t | A) = FA(t) for all continuity points t of FA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' According to the terminology of [66], such a subsequence is called determining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus when investigating asymptotic properties of sufficiently thin subsequences of sequences (Xn)n≥1 with bounded norms, we can assume, without loss of generality, that (Xn)n≥1 itself is determining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As is shown in [27, 66], for any determining sequence (Xn)n≥1 there exists a random measure µ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' a measurable map from the underlying probability space (Ω, F, P) to M) such that for any A with P(A) > 0 and any continuity point t of FA we have (34) FA(t) = EA(µ(−∞, t]) LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 31 where EA denotes conditional expectation given A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This measure µ is called the limit random measure of (Xn)n≥1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see Section 9 below for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' With these preparations, we are now in a position to formulate the subsequence theorems of Aldous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theorem A (Aldous [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a determining sequence with limit ran- dom measure µ and let A be a limit statute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then there exists a subsequence (Xnk)k≥1 such that for any further subsequence (Xmk)k≥1 ⊂ (Xnk)k≥1 we have P((λ(ω), Xm1(ω), Xm2(ω), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') ∈ A) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theorem B (Aldous [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a determining sequence with limit ran- dom measure µ and let T = (f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , {Gλ, λ ∈ M0}) be a weak limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Assume that P(µ ∈ M0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then there exists a sub- sequence (Xnk)k≥1 such that for any further subsequence (Xmk)k≥1 ⊂ (Xnk)k≥1 we have lim k→∞ P(fk(Xm1(ω), Xm2(ω), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' µ(ω)) ≤ t) = EGµ(ω)(t) at all continuity points t of the distribution function on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Writing out Theorem A and B in the case of the limit statutes A1, A2 above and the weak limit theorem defined by (33), we get the CLT, LIL and SLLN for thin subsequences of determining sequences, as stated in (29), (30), (31) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The proof of Koml´os’ result (31) exemplifies the technique used in the field of sub- sequence behavior before Aldous’ paper [27], and in particular in proving the results (29)–(32) mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As Koml´os showed, if (Xn)n≥1 is a sequence of random variables with bounded L1 norms, then its sufficiently thin subsequences (Xnk)k≥1 are, after a random centering and small perturbation, an identically distributed martingale difference sequence with finite means and thus, by classical martingale theory, they satisfy the SLLN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Martingale versions of the CLT and LIL yield also relations (29), (30) and their functional versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' While this method yields several further limit theorems for lacunary sequences, martingale difference sequences cer- tainly do not satisfy all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' limit theorems in a randomized form and thus the general subsequence principle cannot be proved in such a way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The proof of Theo- rems A and B in [27] uses a different way and utilizes near exchangeability properties of subsequences of general sequences of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a deter- mining sequence with limit random measure µ and let (Yn)n≥1 be a sequence of random variables, defined on the same probability space as the Xn’s, conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' with respect to µ, with conditional distribution µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (For the construction of such an (Yn)n≥1 one may need to enlarge the probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') Clearly, (Yn)n≥1 is exchangeable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for any permutation σ : N → N of the positive integers, the sequence (Yσ(n))n≥1 has the same distribution as (Yn)n≥1, and it satisfies limit the- orems of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables in a mixed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, if EY 2 1 < ∞ and 32 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY Y = E(Y1 | µ), Z = Var (Y1 | µ), then N−1/2 N � k=1 (Yk − Y ) D −→ N(0, Z) and lim sup N→∞ (2N log log N)−1/2 N � k=1 (Yk − Y ) = Z1/2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This principle holds in full generality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for all a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' and distributional limit theorems in the above formalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Indeed, if the Yn are conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' with respect to µ and with conditional distribution µ (a random probability measure on R) and if A is a limit statute, then (35) P((µ, Y1, Y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') ∈ A|µ)(ω) = P(µ(ω), Y ∗ 1 , Y ∗ 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') ∈ A) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' where (Y ∗ n )n≥1 is an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' sequence with marginal distribution µ(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By the definition of limit statute, the last probability in (35) equals 1 and taking expectations we get P((µ, Y1, Y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') ∈ A) = 1, which is exactly our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Specializing to the case of the limit statutes A1 and A2 above, we get relations (29) and (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A similar argument works for distributional limit theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Now, as is shown in [27], for every k ≥ 1 we have (36) (Xn1, Xn2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Xnk) D −→ (Y1, Y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , Yk) as n1 < n2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' < nk, n1 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In other words, for large indices the finite dimensional distributions of the sequence (Xnk)k≥1 are close to those of the limiting exchangeable sequence (Yk)k≥1 and thus one may expect that limit theorems of (Yk)k≥1 (which, as we have just seen, are mixed versions of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' limit theorems) continue to hold for sufficiently thin subse- quences (Xnk)k≥1 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Of course, a limit theorem for (Xnk)k≥1 can describe a complicated analytic property of the infinite vector (Xn1, Xn2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , Xnk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') which does not follow from the weak convergence of the finite dimensional distributions of the sequence, but with a suitable thinning procedure and delicate analytic ar- guments, Aldous showed an infinite dimensional extension of (36), leading to the validity of Theorems A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Although the theorems of Aldous are of exceptional generality, there are important results for lacunary sequences which are not covered by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As was shown by Gaposhkin [128],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for every uniformly bounded sequence (Xn)n≥1 of random variables there exists a subsequence (Xnk)k≥1 and bounded random variables X and Y ≥ 0 such that for any numerical sequence (an)n≥1 satisfying (37) AN := N � k=1 a2 k → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' aN = o(A1/2 N ) LACUNARY SEQUENCES IN ANALYSIS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' PROBABILITY AND NUMBER THEORY 33 we have (38) 1 AN N � k=1 ak(Xnk − X) D −→ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' and if the second relation of (37) is replaced by (39) aN = o(AN/(log log AN)1/2) then we have (40) lim sup N→∞ 1 � 2A2 N log log AN N � k=1 ak(Xnk − X) = Y 1/2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The difference of these results from (29) and (30) is that in the CLT and LIL we have weighted sums �N k=1 ak(Xnk −X) instead of ordinary sums �N k=1(Xnk −X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For ev- ery fixed coefficient sequence (an)n≥1 the CLT and LIL in (38) and (39) follow from Theorems A and B, but the subsequence (Xnk)k≥1 provided by the proofs depends on (ak)k≥1 and it is not clear that we can select a subsequence (Xnk)k≥1 satisfying (38) and (39) simultaneously for all considered coefficient sequences (ak)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Another important situation not covered by Aldous’ general theorems is when we investigate permutation-invariance of limit theorems for subsequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Since the asymptotic properties of an exchangeable sequence (Yn)n≥1 do not change after any permutation of its terms, it is natural to expect that the conclusions in Theorem A and B remain valid after an arbitrary permutation of the subsequence (Xnk)k≥1 in the theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, the proofs of Theorem A and B are not permutation- invariant and it does not follow that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', any sequence (Xn)n≥1 of random variables with bounded L1 norms contains a subsequence (Xnk)k≥1 satisfying the strong law of large numbers after any permutation of its terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Using ad hoc methods, the latter result has been proved by Berkes [56] and another classical case, namely the unconditional a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' convergence of series � ck(Xnk − X) under � c2 k < ∞ for subse- quences (Xnk)k≥1 of L2 bounded sequences (Xn)n≥1, has been settled by Koml´os [160] (see [27] for another proof via exchangeability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It clearly would be desirable to pro- vide further general results in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We now formulate some structure theorems for lacunary sequences enabling one to handle problems of the kind discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Recall that if (Xn)n≥1 is a determin- ing sequence with limit random measure µ and (Yn)n≥1 is a sequence conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' with respect to the σ-algebra generated by µ and with conditional marginal distributions µ, then there exists a subsequence (Xnk)k≥1 such that (36) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This shows that, in some sense, for large indices the sequence (Xnk)k≥1 resembles the sequence (Yk)k≥1, but this property is far too weak to deduce limit theorems for (Xnk)k≥1 from those valid for the exchangeable sequence (Yk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The following theorem, proved by Berkes and P´eter [63], shows that with a suitable choice of the subsequence (nk)k≥1, the variables (Xnk)k≥1 can be chosen to be close to the Yk in a pointwise sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We call a sequence (Xn)n≥1 of random variables ε-exchangeable 34 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY if on the same probability space there exists an exchangeable sequence (Yn)n≥1 such that P(|Xn − Yn| ≥ ε) ≤ ε for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then we have Theorem C (Berkes and P´eter [63]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a sequence of random variables bounded in probability, and let (εn)n≥1 be a sequence of positive reals tending to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then, if the underlying probability space is large enough, thee exists a subsequence (Xnk)k≥1 such that, for all l ≥ 1, the sequence Xnl, Xnl+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' is εl-exchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that Theorem C provides a different approximating exchangeable sequence (Y (l) j )j≥1 for each tail sequence (Xnl, Xnl+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' ), with termwise approximating error εl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The following theorem describes precisely the structure of the the sequences (Y (l) j )j≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theorem D (Berkes and P´eter [63]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a determining sequence of random variables, and let (εn)n≥1 be a sequence of positive reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then there exists a subsequence (Xmk)k≥1 and a sequence (Yk)k≥1 of discrete random variables such that (41) P � |Xmk − Yk| ≥ εk � ≤ εk k = 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , and for each k > 1 the atoms of the finite σ-field σ{Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , Yk−1} can be divided into two classes Γ1 and Γ2 such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Firstly, (42) � B∈Γ1 P(B) ≤ εk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Secondly, for any B ∈ Γ2 there exist PB-independent random variables {Z(B) j , j = k, k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' } defined on B with common distribution function FB such that (43) PB � |Yj − Z(B) j | ≥ εk � ≤ εk, j = k, k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here FB denotes the limit distribution of (Xn)n≥1 relative to B and PB denotes conditional probability given B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We now give applications of Theorem D to the problems discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' First we note that using Theorem D it is a simple exercise to prove, for suitable subsequences of a uniformly bounded sequences (Xn)n≥1, the weighted CLT and LIL in (38), (40) simultaneously for all permitted coefficient sequences (an)n≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Next we give a permutation-invariant form of Theorem B for distributional limit theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We call the weak limit theorem T = (f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , S, {Gµ, µ ∈ M0}) regular if there exist sequences pk ≤ qk of positive integers tending to +∞ and a sequence ωk → +∞ such that (i) fk(λ, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') depends only on λ, xpk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , xqk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (ii) fk satisfies the Lipschitz condition |fk(λ, xpk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , xqk) − fk(λ′, x′ pk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , x′ qk)| ≤ ≤ 1 ωk qk � i=pk |xi − x′ i|α + ̺∗(λ, λ′) LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 35 for some 0 < α ≤ 1, where ̺∗ is a metric on M0 generating the same topology as the Prohorov metric ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, the central limit theorem can be formalized by the functions fk(λ, x[k1/4], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , xk) = x[k1/4] + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' + xk − kE(λ) √ k , leading to a regular limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that originally we formalized the CLT with the functions fk in (33) containing all variables x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', but under bounded second moments the first k1/4 terms here are irrelevant and hence we can always switch to the present version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The same procedure applies in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theorem E (Aistleitner, Berkes and Tichy [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a determining sequence with limit random measure ˜µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let T = (f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , S, {Gµ, µ ∈ M0}) be a regular weak limit theorem and assume that P(˜µ ∈ M0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then there exists a subsequence (Xnk)k≥1 such that for any permutation (X∗ k)k≥1 of (Xnk)k≥1 we have (44) fk(X∗ 1, X∗ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , ˜µ) →d � G˜µdP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In case of the CLT formalized above, assuming supn EX2 n < +∞ implies easily that ˜µ has finite variance almost surely, and thus denoting its mean and variance by X and Y , respectively, we see that the integral in (44) is the distribution N(0, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Hence (44) states in the present case that 1 √ N N � k=1 (X∗ k − X) D −→ N(0, Y ), which is the permutation-invariant form of the CLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Concerning a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' limit theorems, a permutation-invariant form of the strong law of large numbers for subsequences of an L1-bounded sequence was proved, as already mentioned, in Berkes [56], and a similar argument yields the corresponding result for the LIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' No permutation-invariant version of the general result in Theorem A has been proved in the literature, but there is no need for that, since a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' limit theorems can be reformulated in a distributional form and thus the proof of Theorem B applies with obvious changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For illustration, we give here the reformulation of the LIL: Theorem F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a sequence of random variables with E|Xn| ≤ 1, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Put Sn = �n i=1 Xi, Sk,l = �l i=k+1 Xi, and L(N) = (2N log log N)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then lim supN→∞ SN/L(N) = 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' iff for any ε > 0 there exists a sequence m1 < m2 < · · of positive integers such that mk ≥ 5k and P � max mk≤j≤mk+1 Sk,j L(j) > 1 + ε � ≤ 2−k, k ≥ k0, and P � max mk≤j≤mk+1 Sk,j L(j) < 1 − ε � ≤ 2−k, k ≥ k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 36 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY It is worth pointing out that given a sequence (Xn)n≥1 of random variables, find- ing a subsequence (Xnk)k≥1 satisfying the permutation-invariant form of some limit theorem generally requires a much faster growing sequence (nk)k≥1 than to find a subsequence to satisfy the original limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This is a phenomenon which also occurs for lacunary trigonometric sums or lacunary sums of dilated functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' com- pare the last paragraph of Section 4 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In conclusion we note that if (Xn)n≥1 is a sequence of random variables with fi- nite means over the probability space (0, 1) equipped with the Borel σ-algebra and Lebesgue measure such that for all n ≥ 1 and (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , an) ∈ Rn we have (45) C1 � n � k=1 |ak|p �1/p ≤ E ����� n � k=1 akXk ����� ≤ C1 � n � k=1 |ak|p �1/p for some p ≥ 1 and positive constants C1, C2, then the closed subspace of L1(0, 1) spanned by the Xn is isomorphic with the ℓp space (Hilbert space if p = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Relation (45) holds, in particular, if the Xn are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' symmetric p-stable random variables with p > 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' their characteristic function (Fourier transform) is given by exp(−c|t|p) with some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus applying the subsequence principle to the “limit theorem” (45) provides important information on the subspace structure of L1(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Using this method, Aldous [28] proved the famous conjecture that every infinite dimensional closed subspace of L1(0, 1) contains an isomorphic copy of ℓp for some 1 ≤ p ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For a further application of this method, see an improvement of the classical theorem of Kadec and Pe�lczy´nski [147] on the subspace structure on Lp, p > 2, in Berkes and Tichy [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' New results: Exact criteria for the central limit theorem for subsequences By the classical resonance theorem of Landau [163], for a real sequence (xn)n≥1 the series �∞ n=1 anxn converges for all (an)n≥1 ∈ ℓp (1 ≤ p ≤ ∞) if and only if (xn)n≥1 ∈ ℓq, where 1/p + 1/q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A deep extension of this result to the case of function series was given by Nikishin [190].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We call a sequence (fn)n≥1 of measurable functions on (0, 1) a convergence system in measure for ℓp if for any real sequence (an)n≥1 ∈ ℓp the series �∞ n=1 anfn converges in measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In the case p = 2 Nikishin proved the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theorem G (Nikishin [190, 191]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A function system (fn)n≥1 over (0, 1) is a conver- gence system in measure for ℓ2 if and only if for any ε > 0 there exists a measurable set Aε ⊂ (0, 1) with measure exceeding 1 − ε and a constant Kε > 0 such that for all N ≥ 1 and all (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , aN) ∈ RN we have (46) � Aε � N � n=1 anfn �2 dx ≤ Kε N � n=1 a2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The sufficiency of (46) is obvious, so the essential (and highly remarkable) state- ment is the converse: if a sequence (fn)n≥1 is a convergence system in measure for LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 37 ℓ2, then, except for a subset of (0, 1) with arbitrary small measure, (fn)n≥1 behaves like an orthonormal sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In the previous section we discussed the subsequence principle stating that suffi- ciently thin subsequences of arbitrary sequences of random variables, subject to mild boundedness conditions, satisfy “all” limit theorems for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables in a mixed (randomized) form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A typical special case of this principle is the following result: Theorem H (Gaposhkin [132]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a sequence of random variables satisfying (47) sup n EX2 n < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then there exists a subsequence (Xnk)k≥1 together with random variables X and Y ≥ 0 such that for any further subsequence (Xmk)k≥1 of (Xnk)k≥1 we have (48) 1 √ N N � k=1 (Xmk − X) D −→ N(0, Y ), where N(0, Y ) denotes the “variance mixture” normal distribution with characteris- tic function E exp(−Y t2/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If (X2 n)n≥1 is uniformly integrable then by well-known compactness results (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [99]) there exist a subsequence (Xmk)k≥1 and random variables X ∈ L2 and Y ∈ L1/2, Y ≥ 0, such that (49) Xmk → X weakly in L2, (Xmk − X)2 → Y 2 weakly in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='6 As Gaposhkin [128] showed, in this case the random variables X, Y in (48) can be chosen as in (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In Theorem H, condition (47) is not necessary: simple examples show (see below) that there exist sequences (Xn)n≥1 of random variables without any finite moments, but having subsequences satisfying (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The purpose of this section is to give nec- essary and sufficient conditions for the existence of subsequences (Xnk)k≥1 satisfying the randomized CLT (48), and it will turn out that our conditions have the same character as Nikishin’s conditions for the existence of a subsequence being a con- vergence system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' “nice” behavior of the sequence on subsets of the probability space with measure as close to 1 as we wish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' To formulate our results, call a sequence (Xn)n≥1 of random variables nontrivial if it has no subsequence converging with positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is easily seen that for non-degenerate sequences the random variable Y in Theorem H is almost surely 6A sequence (ξn)n≥1 of random variables in Lp, p ≥ 1, is said to converge weakly to ξ ∈ Lp if E(ξnη) → E(ξη) for any η ∈ Lq, where 1/p + 1/q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This type of convergence should not be confused with weak convergence of probability measures and distributions, called generally convergence in distribution, and denoted by D −→.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 38 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY positive and Gaposhkin’s theorem can be rewritten in a form involving a pure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' not mixed) Gaussian limit distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theorem J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a nontrivial sequence of random variables satisfying (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then there exists a subsequence (Xnk)k≥1 and random variables X, Y with Y > 0 such that for all subsequences (Xmk)k≥1 of (Xnk)k≥1 and for any set A in the probability space with P(A) > 0 we have (50) PA ��N k=1(Xmk − X) Y √ N < t � → Φ(t) for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here PA denotes the conditional probability with respect to A, and Φ is the cumulative distribution function of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The nontriviality of (Xn)n≥1 is assumed here to avoid degenerate cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If Xnk → X on some set A with positive probability then for any sufficiently thin subsequence (Xmk)k≥1 of (Xnk)k≥1 we have � |Xmk − X| < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' on A, and consequently a−1 N N � k=1 (Xmk − X) → 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' on A for any norming sequence aN → ∞ (random or not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Since for any sequence (Xn)n≥1 satisfying (47) (and in fact any tight sequence (Xn)n≥1) there is a subsequence (Xnk)k≥1 and a measurable partition A ∪ B of the probability space such that Xnk converges on A and is nontrivial on B, there is no loss of generality in assuming that (Xn)n≥1 is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Clearly, if (Xn)n≥1 satisfies the conclusion of Theorem J, then so does the sequence (Xn + 2−nZ)n≥1 for any a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' finite random variable Z, and thus the assumption (47) is, as stated above, not necessary in Theorem J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Below we will give necessary and suf- ficient condition for the CLT for lacunary subsequences of a given sequence (Xn)n≥1 of random variables without any moment assumption on (Xn)n≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' To formulate our results, let us note that if all subsequences (Xmk)k≥1 of a sequence (Xn)n≥1 satisfy (50) for some random variables X, Y , then (Xn)n≥1 is bounded in probability (see Lemma 2 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As mentioned in the previous section, every sequence (Xn)n≥1 of random variables bounded in probability has a subsequence (Xnk)k≥1 which has a limit distribution relative to every set A of the probability space with P(A) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Such a sequence was called determining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This concept is the same as that of stable convergence, introduced by R´enyi [197];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' our terminology follows that of functional analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Hence in our investigations we can assume without loss of generality that the original sequence (Xn)n≥1 is determining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Now if (Xn)n≥1 is determining and FA denotes its limit distribution relative to the set A, then, as we noted in the previous section, there exists a random measure µ (called the limit random measure of (Xn)) such that (51) FA(t) = EA(µ(−∞, t]) LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 39 for any continuity point t of FΩ, where EA denotes conditional expectation relative to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let F• denote the distribution function of µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' we shall call it the limit random distribution of (Xn)n≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We can state now our first new theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a nontrivial sequence of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then the following statements are equivalent: A) There exist a subsequence (Xnk)k≥1 and random variables X, Y with Y > 0 such that (50) holds for all subsequences (Xmk)k≥1 of (Xnk)k≥1 and for any set A ⊂ Ω with P(A) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' B) For every ε > 0 there is a subsequence (Xnk)k≥1 and a set Aε ⊂ Ω with P(Aε) ≥ 1 − ε such that (52) sup k � Aε X2 nkdP < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If (Xn)n≥1 is determining, then two further equivalent statements are: C) We have (53) +∞ � −∞ x2dF•(x) < +∞ almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' D) For every ε > 0 there exists a set Aε ⊂ Ω with P(Aε) ≥ 1 − ε such that (54) +∞ � −∞ x2dFAε(x) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Our second new theorem characterizes sequences (Xn)n≥1 for which (50) holds with X ∈ L2, Y ∈ L1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a nontrivial sequence of random variables defined on an atomless probability space (Ω, F, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then the following statements are equivalent: A) There exists a subsequence (Xnk)k≥1 and random variables X, Y with Y > 0, X ∈ L2, Y ∈ L1/2 such that (50) holds for all subsequences (Xmk)k≥1 of (Xnk)k≥1 and all sets A ⊂ Ω with P(A) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' B) There exists a subsequence (Xnk)k≥1 and sequences (Yk)k≥1, (τk)k≥1 of random variables satisfying (55) Xnk = Yk + τk, where (56) sup k EY 2 k < +∞, � k |τk| < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 40 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY If (Xn)n≥1 has a limit distribution F, then a third equivalent statement is: C) We have (57) +∞ � −∞ x2dF(x) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In other words, for the validity of (50) with X ∈ L2, Y ∈ L1/2, assumption (47) is necessary and sufficient after a small perturbation of (Xn)n≥1, and for identically distributed (Xn)n≥1 even this perturbation is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A particularly simple case when X ∈ L2, Y ∈ L1/2 is satisfied is when X, Y are nonrandom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A trivial example showing the difference between condition (D) of Theorem 1 and condition (C) of Theorem 2 is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let {Hk, k ≥ 1} be a partition of the probability space with P(Hk) = 2−k for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , and let (Xn)n≥1 be a sequence of random variables on this space which is conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' given each Hk with mean 0 and variance 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then (Xn)n≥1 is nontrivial, determining and clearly satisfies condition (D) of Theorem 1, but since it is identically distributed (in fact exchangeable) and since EX2 1 = +∞, condition (C) of Theorem 2 is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Some lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The key for the proof of our theorems is a general structure theorem for lacunary sequences which was proved in [63], and which was stated as Theorem D in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Furthermore, we need the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a sequence of random variables such that for some ran- dom variables X, Y with Y > 0 and for all subsequences (Xnk)k≥1 we have (58) N� k=1 (Xnk − X) Y √ N D −→ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then (Xn)n≥1 is bounded in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Clearly (58) implies that the sequence (XnN − X)/(Y √ N) is bounded in probability as N → ∞, and thus XnN/ √ N is bounded in probability for any sub- sequence (nk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If (Xn)n≥1 were not bounded in probability then one could find a subsequence (mk)k≥1 and a constant c > 0 such that P(|Xmk| ≥ k) ≥ c for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Xmk/ √ k would not be bounded in probability, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a sequence of random variables and assume that for some random variables X and Y > 0 and all sets A ⊂ Ω with P(A) > 0 we have (59) PA \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed N� k=1 (Xk − X) Y √ N < t \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 → Φ(t) for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 41 Assume further that (59) remains valid if we replace X, Y by some random variables X∗ and Y ∗ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then X = X∗ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' and Y = Y ∗ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' From the assumption it follows that the sequences N−1/2 N � k=1 (Xk − X) and N−1/2 N � k=1 (Xk − X∗) are bounded in probability, and thus the same holds for their difference √ N(X−X∗), whence X = X∗ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' To prove Y = Y ∗, fix c > 1 and set A = {Y ∗ ≥ cY }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If P(A) > 0 then clearly we cannot have both (59) and the analogous relation with Y replaced by Y ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus P(A) > 0 for all c > 1 whence Y ∗ ≤ Y a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The same argument yields Y ≤ Y ∗ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , Xn be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables with distribution function F and set Sn = X1 + · · · + Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then for any t > 0 we have (60) P(|Sn| ≤ 2t) ≤ A t √n \uf8eb \uf8ec \uf8ed � |x|≤t x2dF(x) − 2 \uf8eb \uf8ec \uf8ed � |x|≤t xdF(x) \uf8f6 \uf8f7 \uf8f8 2\uf8f6 \uf8f7 \uf8f8 −1/2 , provided the difference on the right-hand side is positive and � |x|≤t dF(x) ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here A is an absolute constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let F ∗ denote the distribution function obtained from F by symmetrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' From a well-known concentration function inequality of Esseen [106, Theorem 2] it follows that the left-hand side of (60) cannot exceed A1 t √n \uf8eb \uf8ec \uf8ed � |x|≤2t x2dF ∗(x) \uf8f6 \uf8f7 \uf8f8 −1/2 , where A1 is an absolute constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Hence to prove (60) it suffices to show that � |x|≤t dF(x) ≥ 1/2 implies (61) � |x|≤2t x2dF ∗(x) ≥ � |x|≤t x2dF(x) − 2 \uf8eb \uf8ec \uf8ed � |x|≤t xdF(x) \uf8f6 \uf8f7 \uf8f8 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let ξ and η be independent random variables with distribution function F, and set C = {|ξ − η| ≤ 2t}, D = {|ξ| ≤ t, |η| ≤ t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then � |x|≤2t x2dF ∗(x) = � C (ξ − η)2dP 42 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY ≥ � D (ξ − η)2dP = 2 � |ξ|≤t ξ2dP · P(|η| ≤ t) − 2 \uf8eb \uf8ec \uf8ed � |ξ|≤t ξdP \uf8f6 \uf8f7 \uf8f8 2 ≥ � |ξ|≤t ξ2dP − 2 \uf8eb \uf8ec \uf8ed � |ξ|≤t ξdP \uf8f6 \uf8f7 \uf8f8 2 , provided P(|η| ≤ t) ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus (61) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Ω, F, P) be an atomless probability space and X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' a se- quence of random variables on (Ω, F, P) with limit distribution F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then there exist a subsequence (Xnk)k≥1 and sequences (Yk)k≥1 and (τk)k≥1 of random variables on (Ω, F, P) such that Xnk = Yk + τk, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , such that the random variables Yk have distribution function F, and such that � k |τk| < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let ( ˆXn)n≥1 be discrete random variables such that P(|Xn − X′ n| ≥ 2−n) ≤ 2−n, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , and denote by Fn the distribution function of Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Clearly Fn → F and thus εn := ̺(Fn, F) → 0, where ̺ denotes the Prohorov distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By a theorem of Strassen [8] there exists a probability measure µn on R2 with marginals Fn and F such that µn((x, y) : |x − y| ≥ εn) ≤ εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let c be a possible value of ˆXn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Since the probability space restricted to A = { ˆXn = c} is atomless, there exists a random variable Vn on this space such that PA(Vn < t) = µn((x, y) : x = c, y < t) µn((x, y) : x = c) for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Carrying out this construction for all possible values of c in the range of ˆXn, we get a random variable Vn defined on the whole probability space such that the joint distribution of ˆXn and Vn is µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Clearly the distribution of Vn is F and P(| ˆXn − Vn| ≥ εn) ≤ εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Choosing (nk)k≥1 so that εnk ≤ 2−k we get P � | ˆXnk − Vnk| ≥ 2−k� ≤ 2−k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' P � |Xnk − Vnk| ≥ 2 · 2−k� ≤ 2 · 2−k and thus � k |Xnk − Vnk| < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' by the Borel–Cantelli lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus the de- composition Xnk = Yk + τk, where Yk = Vnk and τk = Xnk − Vnk, satisfies the requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' □ Our final two lemmas concern the properties of the limit random distribution of determining sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 43 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a determining sequence of random variables with limit random distribution F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then for any set A ⊂ Ω with P(A) > 0 we have (62) EA \uf8eb \uf8ed +∞ � −∞ x2dF•(x) \uf8f6 \uf8f8 = +∞ � −∞ x2dFA(x), in the sense that if one side is finite then the other side is also finite and the two sides are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The statement remains valid if in (62) we replace the interval of integrations by (−t, t), provided t and −t are continuity points of FA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This lemma follows easily from (51) by integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let X, X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' be random variables such that both sequences (Xn)n≥1 and (Xn − X)n≥1 are determining;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' let F• and G• denote, respectively, their limit random distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then +∞ � −∞ x2dG•(x) < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' implies +∞ � −∞ x2dF•(x) < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' and conversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let ε > 0 and choose a set A ⊂ Ω such that P(A) ≥ 1 − ε and on A both X and +∞ � −∞ x2dF•(x) are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let FA and GA denote the limit random distribution of (Xn)n≥1 resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (Xn − X)n≥1 relative to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Replacing Xn by Xn + τn where τn → 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' clearly does not change the limit distributions FA, GA, F•, G•, and thus by passing to a subsequence and using Lemma 4 we can assume, without loss of generality, that the Xn are identically distributed on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then EAX2 1 = EAX2 2 = · · · = +∞ � −∞ x2dFA(x), where the last integral is finite by the boundedness of +∞ � −∞ x2dF•(x) on A and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By Minkowski’s inequality and the boundedness of X on A it follows that EA((Xn − X)2) is also bounded, and thus Fatou’s lemma implies that +∞ � −∞ x2dGA(x) ≤ lim inf n→∞ EA � (Xn − X)2� < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Using Lemma 5 again it follows that +∞ � −∞ x2dG•(x) < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As the measure of A can be chosen arbitrarily close to 1, we get � x2dG•(x) < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' □ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Proof of the theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We begin with the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Using diag- onalization and Chebyshev’s inequality it follows that if a sequence (Xn)n≥1 satisfies (B), then it has a subsequence bounded in probability and thus also a determining subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By Lemma 1 the same conclusion holds if (Xn)n≥1 satisfies (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus 44 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY to prove our theorem it suffices to prove the equivalence of (A), (B), (C), (D) for determining sequences (Xn)n≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In what follows we shall prove the implications (A) =⇒ (C) =⇒ (D) =⇒ (B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' since (B) =⇒ (A) follows easily from Theorem D in the previous section by diagonalization, this will prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Assume that (Xn)n≥1 is a determining sequence satisfying (A), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' there exists a subsequence (Xnk)k≥1 and random variables X, Y with Y > 0 such that for any further subsequence (Xmk)k≥1 of (Xnk)k≥1 and any set A ⊂ Ω with P(A) > 0 we have (63) PA \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed N� k=1 (Xmk − X) Y √ N < t \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 → Φ(t) for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We claim that (Xn)n≥1 satisfies (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Clearly we can assume without loss of generality that (Xnk)k≥1 = (Xk)k≥1 and since (Xn−X)k≥1 contains a determining subsequence, we can assume also that (Xn − X)n≥1 itself is determining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Moreover, since (Xn − X)n≥1 satisfies (C) if and only if (Xn)n≥1 does (see Lemma 6), we can assume that X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Assume indirectly that (Xn)n≥1 does not satisfy (C), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' there exists a set B ⊂ Ω with P(B) > 0 such that (64) lim t→∞ t � −t x2dF•(x) = +∞ on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then there exists a set B∗ ⊂ B with P(B∗) ≥ P(B)/2 such that on B∗ the random variable Y is bounded and (63) holds uniformly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' there exists a constant K > 0 and a numerical sequence Kt → +∞ such that t � −t x2dF•(x) ≥ Kt and Y ≤ K on B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Also, 1 − F•(t) + F•(−t) → 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' as t → ∞, and thus we can choose a set B∗∗ ⊂ B∗ with P(B∗∗) ≥ P(B∗)/2 such that on B∗∗ the last convergence relation holds uniformly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' there exists a positive numerical sequence εt ց 0 such that (65) 1 − F•(t) + F•(−t) ≤ εt on B∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We show that there exists a subsequence (Xmk)k≥1 of (Xnk)k≥1 such that (63) fails for A = B∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Since our argument will involve the sequence (Xn)n≥1 only on the set B∗∗, in the sequel we can assume, without loss of generality, that B∗∗ = Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' That is, we may assume that (65) holds on the whole probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 45 Let C be an arbitrary set in the probability space with P(C) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Integrating (65) and using (51) and Lemma 5 we get (66) t � −t x2dFC(x) ≥ Kt, 1 − FC(t) + FC(−t) ≤ εt, t ∈ H, where H denotes the set of continuity points of FΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Choose t0 ∈ H so large that εt0 ≤ 1/16 and then choose t1 so large that K1/2 t /4 ≥ 2t2 0 for t ≥ t1, t ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then for t ≥ t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' t ∈ H we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' using the second relation of (66),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' ������ t � −t xdFC(x) ������ ≤ 2t2 0 + � t0≤|x|≤t |x|dFC(x) ≤ 2t2 0 + \uf8eb \uf8ec \uf8ed � |x|≥t0 dFC(x) \uf8f6 \uf8f7 \uf8f8 1/2 \uf8eb \uf8ec \uf8ed � |x|≤t x2dFC(x) \uf8f6 \uf8f7 \uf8f8 1/2 ≤ 2t2 0 + 1 4 \uf8eb \uf8ec \uf8ed � |x|≤t x2dFC(x) \uf8f6 \uf8f7 \uf8f8 1/2 ≤ 1 2 \uf8eb \uf8ec \uf8ed � |x|≤t x2dFC(x) \uf8f6 \uf8f7 \uf8f8 1/2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' and thus we proved that for any C ⊂ Ω with P(c) > 0 we have (67) t � −t x2dFC(x) − 2 \uf8eb \uf8ed t � −t xdFC(x) \uf8f6 \uf8f8 2 ≥ 1 2Kt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' t ≥ t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' t ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Since (Xn)n≥1 is bounded in probability, there exists a function ψ(x) ր ∞ such that (68) sup n Eψ(Xn) ≤ 1 (see [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (ak)k≥1 be a sequence of integers tending to +∞ so slowly that ak ≤ log k and (69) δk := ak/ψ(k1/4) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let further (εn)n≥1 tend to 0 so rapidly that εak ≤ 2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By Theorem D there exists a subsequence (Xmk)k≥1 and a sequence (Yk)k≥1 of discrete random variables such 46 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY that (41) holds and for each k > 1 the atoms of the finite σ-field σ{Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , Yak} can be divided into two classes Γ1 and Γ2 such that (70) � B∈Γ1 P(B) ≤ εak ≤ 2−k and for each B ∈ Γ2 there exist i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables Z(B) ak+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , Z(B) k on B with distribution FB such that (71) PB � |Yj − Z(B) j | ≥ 2−k� ≤ 2−k, j = ak + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We show that (72) P \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed N� k=1 Xmk Y √ N < t \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 → Φ(t) for all t cannot hold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' this will complete our indirect proof of (A) =⇒ (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Set S(B) ak,k = k � j=ak+1 Z(B) j , B ∈ Γ2, Sak,k = � B∈Γ2 S(B) ak,k 1B, where 1B denotes the indicator function of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By (71), PB ������ k � j=ak+1 Yj − k � j=ak+1 Z(B) j ����� ≥ 1 � ≤ 2−ak, B ∈ Γ2, and thus using (70) we get (73) P ������ k � j=ak+1 Yj − Sak,k ����� ≥ 1 � ≤ 2 · 2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By (68), (69) and the Markov inequality we have P ������ ak � j=1 Xmj ����� ≥ akk1/4 � ≤ ak sup 1≤j≤ak P � |xmj| ≥ k1/4� ≤ akψ(k1/4)−1 = δk, which, together with (73) and (41), yields (74) P ������ k � j=1 Xmj − Sak,k ����� ≥ 2akk1/4 � ≤ 3 · 2−ak + δk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 47 Applying Lemma 3 to the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' sequence {Z(B) j , ak + 1 ≤ j ≤ k} and using (67) we get PB ������ S(B) ak,k √ k ����� ≤ 1 � ≤ PB � S(B) ak,k √k − ak ≤ 2 � ≤ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' K−1/2 √ k , where the constant is absolute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus using (70) and Y ≤ K it follows that (75) P ����� Sak,k Y √ k ���� ≤ 1 K � ≤ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' K−1/2 √ k + 2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If (72) were true then by (74) and ak ≤ log k we would also have Sk,ak/Y √ k D −→ N(0, 1), which clearly contradicts (75) for large k, since the right-hand side tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This completes the proof of (A) =⇒ (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The remaining implications (C) =⇒ (D) and (D) =⇒ (B) of Theorem 1 are easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Assume first that (C) holds, then for any ε > 0 there exists a set A ⊂ Ω with P(A) ≥ 1 − ε and a constant K = Kε such that +∞ � −∞ x2dF•(x) ≤ K on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Integrating the last relation on A and using Lemma 5 we get (52), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (D) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Assume now that (D) holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for any ε > 0 there exists a set A ⊂ Ω with P(A) ≥ 1 − ε such that (52) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Applying Lemma 4 for (Xn)n≥1 on the set A it follows that there exists a subsequence (Xnk)k≥1 and random variables Yk and τk, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , defined on A such that Xnk = Yk + τk, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , and such that the random variables Yk have distribution FA on A and τk → 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Choose a set B ⊂ A with P(B) ≥ 1 − 2ε such that τk → 0 uniformly on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then clearly (τn)n≥1 is uniformly bounded on B, and further � B Y 2 k dP ≤ � A Y 2 k dP = P(A) +∞ � −∞ x2dFA(x) ≤ +∞ � −∞ x2dFA(x) < +∞ for each k ≥ 1 by the identical distribution of the Yk’s and (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus on B the sequences (Yk)k≥1 and (τk)k≥1 have bounded L2 norms and thus the same holds for 48 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY Xmk = Yk + τk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' sup k � B X2 mkdP < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In view of P(B) ≥ 1 − 2ε this shows that (Xn)n≥1 satisfies statement (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theorem 2 follows from Theorem 1 and a slightly sharper form of Theorems H and J which was proved in [147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We already mentioned the fact that in Theorem H the random variables X, Y appearing in (50) actually satisfy X ∈ L2, Y ∈ L1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Moreover, if instead of (47) we make the slightly stronger assumption that the sequence (X2 n)n≥1 is uniformly integrable then by the weak compactness criteria in L1 and L2 it follows that there exists a subsequence (Xnk) and random variables X ∈ L2, Y ∈ L1/2 such that (76) Xnk → X weakly in L2, (Xnk − X)2 → Y 2 weakly in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As is shown in [163], in this case (50) holds with the random variables X, Y de- termined by (76).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We turn now to the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As in the case of Theorem 1, it suffices to prove the equivalence of statements (A), (B), (C) in the case when (Xn)n≥1 is determining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Also, since replacing Xn by Xn + τn where � |τn| < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' does not affect the validity of (50), the conclusion (B) =⇒ (A) of Theorem (2) is contained in the stronger form of Theorem H mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus it suffices to verify the implications (A) =⇒ (C) and (C) =⇒ (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' To prove (A) =⇒ (C) let us assume that (Xn)n≥1 is determining with limit distribution F, and that there exist a subsequence (Xnk)k≥1 and random variables X ∈ L2, Y ∈ L1/2, Y > 0, such that for all subsequences (Xmk)k≥1 of (Xnk)k≥1 and any set A ⊂ Ω with P(A) > 0 equation (50) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We show +∞ � −∞ x2dF(x) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Clearly we can assume without loss of generality that (Xnk)k≥1 < (Xk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Fix ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By the implication (A) =⇒ (B) =⇒ (D) of Theorem 1 there is a set A ⊂ Ω with P(A) ≥ 1 − ε and a subsequence (Xnk)k≥1 such that (77) sup k � A X2 nkdP < +∞ and � A x2dFA(x) < +∞, where FA is the limit distribution of (Xn)n≥1 on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Applying Lemma 4 for (Xnk)k≥1 on A it follows that there exists a subsequence (Xmk)k≥1 of (Xnk)k≥1 admitting the decomposition (78) Xmk = Yk + τk on A, where the Yk are identically distributed on A with distribution function FA and � |τk| < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Being an identically distributed sequence with finite expec- tation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' the sequence (Y 2 k )k≥1 is uniformly integrable on A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' and thus by the sharper form of Theorem H mentioned above it follows that there exists a subsequence (Ypk)k≥1 of (Yk)k≥1 such that Ypk → U weakly in L2(A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (Ypk − U)2 → V 2 weakly in L1(A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' LACUNARY SEQUENCES IN ANALYSIS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' PROBABILITY AND NUMBER THEORY 49 and for any B ⊂ A with P(B) > 0 we have PB ��N k=1(Ypk − U) V √ N < t � → Φ(t) for all t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' where U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' V are random variables such that U ∈ L2(A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' V ∈ L1/2(A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' V > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus by (78) and � |τk| < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' on A we get PB ��N k=1(Xmpk − U) V √ N < t � → Φ(t) for all t for any B ⊂ A with P(B) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Comparing with (50) and using Lemma 2 we get U = X, V = Y a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' on A, and thus we proved that Ypk → X weakly in L2(A), (Ypk − X)2 → Y 2 weakly in L1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Hence EAY 2 = lim k→∞ EA(Ypk − X)2 = lim k→∞(EAY 2 pk − 2EAYpkX + EAX2) = lim k→∞ EAY 2 pk − EAX2 = +∞ � −∞ x2dFA(x) − EAX2, (79) where in the last step we used the fact that the Yk’s have distribution FA on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Hence (80) P(A)−1 � A Y 2dP = +∞ � −∞ x2dFA(x) − P(A)−1 � A X2dP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Since X ∈ L2(Ω), Y 2 ∈ L1(Ω), the left-hand side of (80) and the second term on the right-hand side approach finite limits as P(A) → 1 and thus � +∞ −∞ x2dFA(x) also converges to a finite limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the other hand, FA → F as P(A) → 1 and thus Fatou’s lemma implies +∞ � −∞ x2dF(x) ≤ lim inf P (A)→1 +∞ � −∞ x2dFA(x) < +∞, proving the implication (A) =⇒ (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Now if (C) holds then by Lemma 4 there exists a subsequence (Xnk)k≥1 permitting the decomposition (55) where � |τk| < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' and Yk are identically distributed with distribution F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' since F has finite variance by (C), the first relation of (56) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus (Xn)n≥1 satisfies (B) and the proof of Theorem 2 is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Acknowledgments Christoph Aistleitner is supported by the Austrian Science Fund (FWF), projects F- 5512, I-3466, I-4945, I-5554, P-34763, P-35322 and Y-901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Istvan Berkes is supported by Hungarian Foundation NKFI-EPR No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' K-125569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 50 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Adler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Keane, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Smorodinsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A construction of a normal number for the con- tinued fraction transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Number Theory, 13(1):95–105, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Airey and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Normality of different orders for Cantor series expansions.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Berkes, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Tichy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Lacunary sequences and permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In Depen- dence in probability, analysis and number theory, pages 35–49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Kendrick Press, Heber City, UT, 2010.' metadata={'source': 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Tracts in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Cambridge University Press, Cambridge, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [76] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Bugeaud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the expansions of a real number to several integer bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Mat.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 9:235–245, 1969/70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [82] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Chatterji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Un principe de sous-suites dans la th´eorie des probabilit´es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In S´eminaire de Probabilit´es, VI (Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Strasbourg, ann´ee universitaire 1970-1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Journ´ees Probabilistes de Strasbourg, 1971), pages 72–89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Lecture Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 258.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Diffusive behavior of ergodic sums over rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 19(2):1950016, 26, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [87] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='-P.' metadata={'source': 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Mathematics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Interscience Publishers, 1958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [100] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Erd˝os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the strong law of large numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 67:51–56, 1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [101] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Erd˝os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the convergence of trigonometric series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 22:37– 39, 1943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [102] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Erd˝os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On trigonometric sums with gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Magyar Tud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Kutat´o Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' K¨ozl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 7:37–42, 1962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 54 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY [103] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Erd¨os and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' G´al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the law of the iterated logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' I, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Nederl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Wetensch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 58 = Indag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Wetensch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 52:851–854 = Indagationes Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 11, 299–302 (1949), 1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [105] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Monatsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 199(1):113-133, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [112] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Fukuyama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The central limit theorem for Riesz-Raikov sums.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Hungar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 118(1-2):155–170, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [114] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Fukuyama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The law of the iterated logarithm for the discrepancies of a permutation of {nkx}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Hungar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 123(1-2):121–125, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [115] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Fukuyama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A central limit theorem and metric discrepancy result for sequences with bounded gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Dependence in Probability, Analysis and Number Theory, pp.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 180(4):731–742, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [126] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Furstenberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Math.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (2), 47:33–49, 1946.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [146] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Kac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Probability methods in some problems of analysis and number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Amer.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Bases, lacunary sequences and complemented subspaces in the spaces Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Studia Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 21:161–176, 1961/62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [148] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Kaufman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the approximation of lacunary series by Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Acta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Hung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 35(1-2): 61–66 , 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [149] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Kesseb¨ohmer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Munday, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Stratmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Infinite ergodic theory of numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' De Gruyter, Berlin, 2016.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 56 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY [156] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Koksma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Paris 349:1261–1263, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [221] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Weiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The law of the iterated logarithm for lacunary trigonometric series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Trans.' metadata={'source': 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a/KtAyT4oBgHgl3EQff_jL/content/tmp_files/2301.00352v1.pdf.txt b/KtAyT4oBgHgl3EQff_jL/content/tmp_files/2301.00352v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..389f2a3282cca052c2d88c3ee6aeb5895efdf500 --- /dev/null +++ b/KtAyT4oBgHgl3EQff_jL/content/tmp_files/2301.00352v1.pdf.txt @@ -0,0 +1,992 @@ + + + + + +Manuscript Template + Page 1 of 22 + +FRONT MATTER + +Title +• Controlling Electromagnetic Surface Waves with Conformal Transformation Optics + +Authors +Xiaoyu Zhao,1† Hong Deng,1† Xiaoke Gao,1† Xikui Ma,1 Tianyu Dong1* +1School of Electrical Engineering, Xi’an Jiaotong University, Xi’an 710049, China. + +†These authors contributed equally to this work. +*To whom correspondence should be addressed; E-mail: tydong@mail.xjtu.edu.cn. + +Abstract +The application of transformation optics to the development of intriguing electromagnetic +devices can produce weakly anisotropic or isotropic media with the assistance of quasi- +conformal and/or conformal mapping, as opposed to the strongly anisotropic media +produced by general mappings; however, it is typically limited to two-dimensional +applications. By addressing the conformal mapping between two manifolds embedded in +three-dimensional space, we demonstrate that electromagnetic surface waves can be +controlled without introducing singularity and anisotropy into the device parameters. +Using fruitful surface conformal parameterization methods, a near-perfect conformal +mapping between smooth manifolds with arbitrary boundaries can be obtained. +Illustrations of cloaking and illusions, including surface Luneburg and Eaton lenses and +black holes for surface waves, are provided. Our work brings the manipulation of surface +waves at microwave and optical wavelengths one step closer. + +Teaser +Waves can be controlled at will on arbitrary open surfaces without holes, showing +fascinating applications such as invisible bumps for surface waves, reproducing +scatterings of one bump on other smooth surfaces, and controlling light beams on surfaces +to focus, to bend and/or to be absorbed akin to black holes without visible scatterings. + +MAIN TEXT + +Introduction +Since its inception in the design of electromagnetic cloaks (1, 2), transformation optics +(TO) has proven to be a powerful tool for understanding and customizing the physics in +acoustics (3), optics (4), mechanics (5), thermodynamics (6, 7), etc. Following the +groundbreaking work of cloaking, a number of other electromagnetic devices have been +reported within the theoretical framework of TO, such as electromagnetic concentrators +(8, 9), field rotators (10), optical lenses (11, 12) and optical illusion devices (13, 14). In +practice, however, traditional TO often yields significant anisotropy in a designed medium +(15). Thus, metamaterials are often used to infer spatial changes from coordinate +transformation geometry, which is based on the mathematical equivalence between +geometry and material (16). + +Manuscript Template +Page 2 of 22 + +To reduce the anisotropy of the functional medium induced by TO, various approaches +have been developed. By constructing mapping in non-Euclidean space, for instance, it is +possible to remove singular points formed by traditional TO (17), hence minimizing +anisotropy in part. But for wavelengths comparable to the size of the transform region, +non-Euclidean TO may perform even worse (18); thus, several research projects focus on +conformal or quasi-conformal mappings to achieve isotropy (19). In ℝ2, the concept of a +carpet cloak that resembles a flat ground plane is successfully realized with an isotropic +medium produced by minimizing the Modified-Liao functional under sliding boundary +conditions (20), or equivalently by constructing the quasi-conformal mapping via solving +inverse Laplace's equations (21). Although the concept of carpet cloak has been extended +to ℝ3 by the extrusion or revolution of a two-dimensional refractive index profile to +control the reflection of free-space waves, it is only applicable to surfaces with +translational or rotational symmetry (22). +Previous research has focused largely on controlling propagating waves by TO, whereas +less attention has been attached to the manipulation of surface waves (12, 23, 24). Perfect +surface wave cloaks have been proposed by equating the optical path length of a ray +traversing a flat plane with a homogeneous refractive index to the optical path on a curved +surface with an angle-dependent refractive index for two orthogonal paths (25, 26), which +have been experimentally validated (27). Although an electrically large object may be +hidden by such a cloak with an inhomogeneous isotropic medium, this approach is limited +to rotationally symmetric surfaces. By linking the governing eikonal equations on a virtual +flat plane and on a curved surface by transformation optics, the projection mapping yields +surface wave cloaks for non-rotationally symmetric geometries but with high anisotropy +(14, 28). Considerable effort has been devoted to reducing such anisotropy by employing +efficient numerical conformal algorithms such as boundary first flattening (29), yet only +non-rotationally symmetric surfaces with circular boundary are investigated (30). +In this work, we show how to manipulate surface waves on smooth manifolds embedded +in ℝ3 within the framework of conformal TO, requiring an effective isotropic material +under the regime of geometrical optics. Fig. 1 illustrates a conformal surface mapping +between two smooth manifolds in ℝ2 and ℝ3, i.e., 𝑓: ℳ′ → ℳ. The curved manifold ℳ +shown in Fig. 1A has been 𝑢𝑣-parameterized and the mesh grid can be regarded as the +mapping result of the Cartesian coordinate system {𝑥′, 𝑦′} in Fig. 1B. When the mapping +is conformal or quasi-conformal, the face element d𝑆 remains right-angled, indicating that +elements are just scaled up with little distortion. From the local coordinate systems on d𝑆 +and d𝑆′ (Fig. S5), one can derive the Jacobian matrix 𝐉 of mapping 𝑓 with two singular +values 𝜎J1 = 𝜎J2 = 𝜎J that state equal scaling in two orthogonal directions (31). +Consequently, an isotropic cloaking medium distribution 𝑛 = 1/√det(𝐉) = 1/𝜎J may be +obtained based on the conformal TO (19), representing the ratio of line element d𝑙′ in +virtual space to the scaled element d𝑙 in physical space for compensating optical path +length (2). As a result, light propagating on curved ℳ behaves as propagating on flat ℳ′. +In practice, it is more convenient to describe mesh vertices in ℝ3 in a Cartesian coordinate +system {𝑥, 𝑦, 𝑧} and the Jacobian derived from the local coordinate system forms an +asymmetric rank-two matrix 𝐉3×2. In addition, the possible quasi-conformal mappings can +be measured by the conformality, i.e., the ratio 𝑄 = max(𝜎J1/𝜎J2, 𝜎J2/𝜎J1). A unity ratio +𝑄 allows an effective cloaking medium expressed as 𝑛cloak = 1/√𝜎J1𝜎J2 for every face +element (20). + +Manuscript Template +Page 3 of 22 + + +Results +Having obtained a conformal mapping between the manifolds ℳ ∈ ℝ3 and ℳ′ ∈ ℝ2, we +first design an isotropic surface wave cloak under the perspective of conformal TO and +compare its performance with the traditional surface wave cloak with anisotropic medium +(14). Simulations were conducted on a double-camelback bump with an elliptical base +profile embedded in ℝ3, as shown in Fig. 2. In comparison with the scattering when the +surface has no index profile (Fig. S1A), one can observe that the surface wave cloaking is +successfully achieved by two distinct approaches: one induced by the projection mapping +proposed in (14) (Fig. 2A) and the other originated from the proposed quasi-conformal +mapping (Fig. 2B). The corresponding material characteristics for the two types of cloaks +are displayed in Fig. 2C, indicating that the former is strongly anisotropic while the latter +is almost isotropic. In addition, the isotropic refractive index 𝑛c,double (the subscript "c" +denotes the cloak, and "double" denotes the double-camelback bump) ranges from 0.83 to +1, which decreases as the bump height rises because a longer geometrical distance need to +be compensated by a smaller refractive index in order to attain equal optical path length. +The proposed scheme based on conformal TO has achieved near-perfect surface wave +cloaking while eliminating the anisotropy in the transformation medium that the +traditional scheme presents. The distribution of 𝑛c,double in Fig. 2C outlines an +asymmetric geometric profile, manifesting that the effectiveness of this scheme is +independent from any symmetry. Such an achievement demands mappings with high +conformality rather than those bringing large distortion such as the projection mapping +(14). The numerical method we adopt here (29) can obtain a quasi-conformal mapping +with 𝑄 < 1.03, as shown in Fig. S1B, which is sufficient for designing an effective +isotropic cloaking medium distribution. +As the antithesis of cloaking, optical illusion devices can reproduce the scattering +characteristics of a specific object on other objects through a transformation medium (13, +14). Fig. 3A depicts the surface electromagnetic wave scattered by a single-camelback +bump ℳ filled with homogeneous material. Traditionally, if one wants to reproduce its +scattering on a plane region ℳ′, the quasi-conformal mapping for designing the illusion +device is 𝑓′: ℳ → ℳ′ with a Jacobian matrix 𝚲2×3. Fig. 3B shows the accurately +recurring scattering characteristics on plane region ℳ′ filled with 𝑛i,plane = 1/√𝜎Λ1𝜎Λ2 +(the subscript "i" denotes the illusion, and "plane" denotes the plane region), where 𝜎Λ1 +and 𝜎Λ2 are singular values of 𝚲2×3. Furthermore, Fig. 3C illustrates that the double- +camelback bump filled with a carefully designed isotropic medium distribution can +reproduce the same scattering pattern as shown in Fig. 3A. Such an illusion is realized by +cascading two conformal mappings described in Fig. S3, i.e., 𝑓1 from ℝ3 (virtual space) to +ℝ2 (intermediate space), and 𝑓2 from ℝ2 to the ℝ3 (physical space). Thus, the illusion +medium for the double-camelback bump reads 𝑛i,double = 𝑛i,plane ∙ 𝑛c,double. Fig. 3D +displays the profiles of 𝑛i,plane (for Fig. 3B) and 𝑛i,double (for Fig. 3C), respectively, +which range from 1 to 1.25 (𝑛i,plane) and from 0.85 to 1.21 (𝑛i,double). +The scattering pattern of the single-camelback bump (Fig. 3A) has been successfully +reproduced on the plane region (Fig. 3B) and on the double-camelback bump (Fig. 3C), +which demonstrates that the proposed scheme is a general solution to illusion design on +smooth two-dimensional manifolds. The cascading method to construct mappings between +manifolds embedded in ℝ3 can even tackle surfaces with different base profiles, since a + +Manuscript Template +Page 4 of 22 + +conformal mapping between simply-connected regions in ℝ2 exists according to the +Riemann mapping theorem (32). Moreover, the quasi-conformal ratios 𝑄 of the two +mappings for the double-camelback and single-camelback bump are smaller than 1.03 +(Fig. S1B) and 1.012 (Fig. S2C), respectively, implicating that the cascaded mapping +meets the requirement for high conformality. The range of 𝑛i,single (1 to 1.25) is the +inverse of that of the cloaking refractive index 𝑛c,single (0.8 to 1) shown in Fig. S2B, +because the illusion can be regarded as the inverse design of the cloaking such that the +Jacobian matrices of their corresponding mappings are the Moore–Penrose pseudo-inverse +of each other (31). +Now that the wave behavior on the curved manifold can be manipulated flexibly, it is +natural to consider designing various complicated devices on it, such as surface wave +Luneburg lens, Eaton lens and black hole for surface waves (12, 23, 33, 34). Traditional +designs are usually based on spherical or circular profiles with a constant radius. While for +an elliptical profile without a constant radius, we adopt the distance from the point on the +ellipse to the center, also the coordinate origin, as the generalized radius, i.e., 𝑅(𝜃) = +√(𝑎 cos 𝜃)2 + (𝑏 sin 𝜃)2 (35–37). Thus, the refractive index of the considered Luneburg +lens can be expressed as + +𝑛L(𝑟, 𝜃) = √2 − (𝑟/𝑅(𝜃))2, +(1) +where 𝑟 = √𝑥2 + 𝑦2 and 𝜃 = arctan(𝑦/𝑥). Similar to the traditional circular Luneburg +lens, such a distribution retains 𝑛L = 1 on the boundary and 𝑛L = √2 at the center 𝑟 = 0 +(38). Next, the medium distribution for a Luneburg lens on the double-camelback bump +can be expressed as 𝑛Luneburg = 𝑛c,double ⋅ 𝑛L. As illustrated in Fig. 4A, two Gaussian +beams with a free-space wavelength 𝜆G = 50 mm are incident along the 𝑥-direction at the +position ±0.8𝑏 on the 𝑦-direction and reflected by the Luneburg lens to interfere at the +focus point. The focal distance reads 20𝜆G that is identical to the unit circular Luneburg +lens. For the Eaton lens, the refractive index 𝑛E reads as + +𝑛E(𝑟, 𝜃) = √2𝑅(𝜃)/𝑟 − 1, +(2) +which can approach infinity when 𝑟 = 0, leaving a singular point to be cared for. Fig. 4B +describes that a Gaussian beam going along the 𝑥-direction bends to the inverse 𝑥- +direction after passing through the Eaton lens on the double-camelback bump. The +proposed surface wave Luneburg and Eaton lenses may be deployed in optical imaging, +signal acquisition and novel designs for surface wave microwave antennas. Another +functional device that can rotate beam propagation is the peripheral of the two-layer +optical black hole, where light is compelled to travel in a spiral path into the absorbing +medium at the core. The piece-wise refractive index distribution function 𝑛B can be +expressed as + +𝑛B(𝑟, 𝜃) = { + 1, + 𝑟 > 𝑅(𝜃) + 𝑅(𝜃)/𝑟, + 𝑟𝑐 ⋅ 𝑅(𝜃) < 𝑟 < 𝑅(𝜃) + 1/𝑟𝑐 + i𝛾, + 𝑟 < 𝑟𝑐 ⋅ 𝑅(𝜃) +, +(3) +where 𝑟𝑐 = 0.4 is the scaling factor of the internal ellipse core compared with the base +profile and 𝛾 = 0.1 is the loss factor. The refractive index distribution 𝑛Blackhole = +𝑛c,double ⋅ 𝑛B on the double-camelback bump is depicted in Fig. 4D. The real part of +material parameters is matched on the inner boundary, and the imaginary part for +absorbing energy ranging from 0.083 to 0.097 only exists in the core. The same Gaussian + +Manuscript Template +Page 5 of 22 + +beam that was used for the Eaton lens is employed, and the result in Fig. 4C shows that +the beam bends around 90∘ before it reaches the inner boundary and is absorbed by the +lossy core without reflection, showing potential application in interference reduction and +energy harvesting for electronic devices. Note that, the overall sizes of the simulation +models are larger than ten times the operating wavelength, demonstrating that the +proposed scheme is capable of managing surface wave behaviors on electrically large +objects. Moreover, the excellent performance of these functional surface wave devices +demonstrates that, based on the proposed scheme, a variety of novel devices may be +realized on smooth curved manifolds, which may facilitate the development of +miniaturized and integrated photonic devices. + +Discussion +Our theory and method are based on geometrical optics. It requires small curvature and +little variation in wavelength (see (7) and (8) in Materials and Methods), which can be +expressed as + +𝑤 = |∇𝜆| = |∇(𝜆0/𝑛)| = 𝜆0|∇𝑛|/𝑛2 ≪ 1, +(4) + +𝜌 = |𝑅𝑖𝑗|𝜆2 = |𝐾𝑔𝑖𝑗|(𝜆0/𝑛)2 = det(𝑔𝑖𝑗) 𝐾2𝜆0 +2/𝑛2 = 𝐾2𝜆0 +2/𝑛6 ≪ 1, +(5) +where 𝑅𝑖𝑗 is the Ricci curvature tensor, 𝐾 is the Gaussian curvature, and 𝑔𝑖𝑗 is the metric +tensor. Both the wavelength index 𝑤 and the curvature index 𝜌 are inversely proportional +to powers of the refractive index 𝑛. In order to prevent 𝑤 and 𝜌 from increasing +drastically, a height lower than half of the base radius is favorable, and thereby the optical +path length can be compensated with a near-unity refractive index. On this basis, +requirements (4) and (5) demand shorter wavelength 𝜆0 and smoother geometric structure +to ease the changing rate |∇𝑛| and the Gaussian curvature 𝐾. As a negative example, a +hemisphere surface wave cloak is reviewed and results are displayed in Fig. S4, whose +refractive index 𝑛c,sphere is between 0.5 and 1 and the maximum of quasi-conformal ratio +𝑄 is smaller than 1.012. The visible scattering appearing in Fig. S4C implies the failure of +geometrical optics because of the high curvature index 𝜌 > 20 residing in the right-angle +connection between the hemisphere and the plane, as is depicted in Fig. S4F, and the +average curvature index 𝜌̅ = 1.57 is also larger than 1. The non-smooth connection causes +the phase distortion in the backward scattering, and the maximum of the forward +scattering |𝐸𝑧 − 𝐸b𝑧|max = 0.75 V/m implies a phase difference arcsin (0.75) = 48.6∘ +resulted from the reconstruction of wave fronts. In comparison, Fig. S1C and Fig. S2D +display the average curvature index 𝜌̅ = 0.54 for double-camelback bump and 𝜌̅ = 0.39 +for single-camelback bump, respectively, both satisfying the requirement (5) and leaving +near-zero 𝜌 on smooth boundaries. One may notice that the wavelength index 𝑤 for the +cloaks shown in Fig. S1D, Fig. S2E and Fig. S4E is smaller than unity everywhere +because it is related to lower powers of 𝜆0 and 𝑛; thus, it is much easier to meet the +requirement of (4) compared to (5). These selected curvature and wavelength +characteristics that validate the approximation of geometrical optics are indispensable for +the excellent performance of electromagnetic devices. + +The isotropic case that determines the expression of requirements (4) and (5) is based on +the conformal or quasi-conformal mappings between two-dimensional manifolds. +Benefiting from the rapid development in conformal parameterization, a series of mapping +methods can be employed to design surface wave carpet cloak (29, 39, 40). The boundary +first flattening (BFF) method (29) adopted in our study can establish near-perfect +conformal mappings not only between smooth manifolds but also surfaces with cuspidal + +Manuscript Template +Page 6 of 22 + +points, such as sharp corners and cone singularities, offering exhilarating promise for +wave manipulation on more complicated surfaces. In addition, there are algorithms aimed +at constructing quasi-conformal mappings between high-genus manifolds (41, 42), which +can be used to deal with phase regulation on surfaces with holes. One noteworthy idea is +to map a high-genus surface to a zero-genus plane region by transforming holes to slits +(43, 44) that implies the possibility for the scheme conducted in simply-connected regions +to manipulate wave behaviors on multiply-connected surfaces. By reasonably utilizing +advanced algorithms for a variety of particular cases, our method has the potential to be a +universal scheme for controlling surface electromagnetic waves on an arbitrary two- +dimensional manifold. + +In summary, we have proposed a general method to manipulate electromagnetic waves on +smooth two-dimensional manifolds without rotational symmetry by means of a certain +isotropic refractive index distribution derived from the quasi-conformal mapping. The +relationship between medium and mappings is induced from the wave equation on the +manifold under the geometrical optics approximation. Numerical quasi-conformal +algorithms are introduced to construct mappings between manifolds, and consequent +functional mediums are validated by cloaking surfaces and generating illusions on plane +regions. By cascading mappings between ℝ2 and ℝ3 to obtain a mapping between ℝ3, we +succeed in reproducing the scattering of a surface on another surface. In addition, +functional devices such as surface Luneburg lenses, surface Eaton lenses, and black holes +for surface waves are designed based on carpet cloaks. Finally, the indices required by +geometrical optics are reviewed to demonstrate the validity of the approximation on +simulation models. Our method paves the way for the regulation of surface +electromagnetic waves on any two-dimensional manifold, and can be utilized to control +surface waves in other fields, such as acoustics, mechanics, and thermodynamics. + +Materials and Methods +Conformal transformation optics for surface waves +Wave equation on curved manifold. The concept of transformation medium stems from +the equivalence between geometry and media. Within the Einstein summation convention, +the Maxwell's wave equation for the electric field ∇ℳ × ∇ℳ × 𝐄 − 𝜇0𝜀0𝜕𝑡 +2𝐄 = 0 in free +space can be expressed as (16) + +𝛻𝑗𝛻𝑗𝐸𝑖 − 𝑅𝑖𝑗𝐸𝑗 − 𝑐0 +−2𝜕𝑡 +2𝐸𝑖 = 0, +(6) +where 𝑐0 = 1/√𝜇0𝜀0 is the light velocity in free space; 𝑅𝑖𝑗 is the Ricci tensor of the +considered geometry ℳ. Supposing that the electromagnetic waves are confined nearby a +curved surface ℳ embedded in ℝ3 as surface waves, its local plane wave solution reads +as 𝐸𝑖 = ℰi𝑒i𝜑 with constant complex amplitudes ℰi, where the phase reads as 𝜑 = k ⋅ r − +𝜔𝑡 with the wave vector 𝐤 = ∇ℳ𝜑 and angular frequency 𝜔 = −𝜕𝑡𝜑. For surface waves, +the wave vector k lies in the tangent space of the curved surface ℳ, i.e., 𝐤 ∈ 𝒯(ℳ). +Thus, (6) can be simplified and approximated in the regime of geometrical optics where +the wavelength 𝜆 = 2𝜋/𝑘 varies slowly with distance, i.e., + +|∇ℳ𝜆| ≪ 1. +(7) +In addition, the effective curvature of the curved surface should be small enough +compared to the wavelength so that the assumption of locally plane waves is valid, i.e., + +|𝑅𝑖𝑗|𝜆2 ≪ 1. +(8) + +Manuscript Template +Page 7 of 22 + +As a result, inserting 𝐸𝑖 = ℰi𝑒i𝜑 into (6) and considering that the (spatial and temporal) +derivatives of ℰi vanish, one can obtain the dispersion relation for the surface wave +propagating on ℳ, which reads as + +𝑘2 = 𝑘𝑗𝑘𝑗 = 𝑔𝑖𝑗𝑘𝑖𝑘𝑗 = 𝜔2/𝑐0 +2. +(9) +Here, 𝑔𝑖𝑗 is the induced metric tensor for the curved surface ℳ, which can be determined +from the transformation Jacobian matrix from the manifold ℳ′ in ℝ2 to ℳ (31),. + +Wave equation on a flat plane. Alternatively, if ℳ is flat (i.e., 𝑅𝑖𝑗 = 0) and filled with +anisotropic medium denoted by relative permeability tensor 𝜇𝑖𝑗, (6) becomes + +∇ × ∇ × 𝐄 − 𝜇0𝜀0𝝁 ⋅ 𝜕𝑡 +2𝐄 = 0. +(10) +Suppose that the electromagnetic waves are confined nearby ℳ and the electric field 𝐄 is +perpendicularly polarized. In a Cartesian coordinate system, if ℳ can be placed into 𝑥𝑦 +plane, we focus on the case that the electric field vector 𝐄 lies in the normal space of the +flat plane ℳ, i.e., 𝐄 ∈ 𝒩(ℳ), and the global wave solution may read as 𝐸𝑧 = ℰ𝑧𝑒i𝜑. +Thus, the phase 𝜑 is independent of 𝑧 and the wave vector just lies on the plane as 𝐤 = +(𝑘𝑥, 𝑘𝑦, 0), because a flat plane is coincident with its tangent space. Since the flat +manifold ℳ has a zero-curvature tensor, the condition (7) holds naturally. Once the other +condition (8) that wavelength varies slowly is satisfied, one may disregard the derivatives +of complex amplitude after inserting 𝐸𝑧 = ℰ𝑧𝑒i𝜑 into (10) and obtain the dispersion +relation for the surface wave propagating on ℳ, which reads as (𝜇𝑥𝑥𝑘𝑥2 + 2𝜇𝑥𝑦𝑘𝑥𝑘𝑦 + +𝜇𝑦𝑦𝑘𝑦2)/det(𝝁) = 𝜔2/𝑐0 +2. By excluding consideration of the particular polarization, the +dispersion equation can be recast within the Einstein summation convention as + +1 +det(𝝁) 𝜇𝑖𝑗𝑘𝑖𝑘𝑗 = +𝜔2 +𝑐0 +2. +(11) + +Transformation medium and geometry. For electromagnetic waves that behave +identically on two manifolds, one can obtain the equivalence between geometry and +material properties by comparing (9) and (11), which yields + +𝜇𝑖𝑗 +det(𝝁) = 𝑔𝑖𝑗. +(12) +The relative permeability tensor 𝜇𝑖𝑗 actually creates an illusion on the flat plane because a +spatial point filled with medium 𝝁 is equivalent to be with a metric 𝒈 = det(𝝁)𝝁−1. If the +local Cartesian coordinate system at this point is aligned along the orthogonal +eigenvectors of 𝝁, the real and symmetric permeability tensor will reduce to +diag(𝜇𝑥, 𝜇𝑦, 𝜇𝑧) so that the square of the line element on 𝑥 direction is d𝑠2 = 𝑔𝑥𝑥d𝑥2 = +𝜇𝑦𝜇𝑧d𝑥2, which is also the square of optical path length in curved free space. In +comparison to d𝑠2 = 𝑛𝑥2d𝑥2 on the flat manifold, one can derive 𝑛𝑥2 = 𝜇𝑦𝜇𝑧 and similar +results on 𝑦 and 𝑧 directions. Consequently, the relationship between the relative +permeability tensor 𝝁 and the refractive index tensor 𝒏 may be expressed as 𝒏2 = +det(𝝁)𝝁−1 and one may further obtain + +𝒏illustion +2 += 𝒈. +(13) +by referring to (12). + + +Manuscript Template +Page 8 of 22 + +Surface transformation and TO medium. The metric tensor in equation (13) is induced +from the mapping 𝑓: ℳ′ → ℳ and can be constructed by the Jacobian matrix 𝐉3×2 as 𝒈 = +𝐉T𝐉 (31). Nevertheless, we prefer to associate 𝒏illusion with the Jacobian matrix 𝚲2×3 that +represents the transformation from ℝ3 (virtual space) to ℝ2 (physical space). Actually, the +asymmetric Jacobian matrices 𝐉3×2 and 𝚲2×3 can be denoted as the Moore–Penrose +pseudo-inverse of each other (31), i.e., 𝐉 = 𝚲†, where the superscript '†' denotes pseudo- +inverse. Thus, one can rewrite the equivalence (13) as + +𝒏illustion +2 += 𝒈 = 𝐉T𝐉 = (𝚲𝚲T)−1. +(14) +Similar relationship can be obtained for cloaking medium 𝒏cloak and corresponding +Jacobian matrix 𝐉3×2 from ℝ2 (virtual space) to ℝ3 (physical space) as + +𝒏cloak +2 += (𝐉T𝐉)−1. +(15) +For the mapping between ℝ3 (Fig. S3), which is formed by cascading two transformations +between ℝ3 and ℝ2, the consequent medium for the illusion can be recast as the +combination of the cloaking and illusion refractive index tensors, i.e., + +𝒏illustion +2 += (𝚲1𝚲1 +T)−1 ⋅ (𝐉2 +T𝐉2)−1. +(16) +where 𝚲1 and 𝐉2 are Jacobian matrices for mappings 𝑓1 and 𝑓2, as illustrated in Fig. S3, +respectively. In particular, when the mappings are conformal, the refractive index +becomes isotropic, and the corresponding Jacobian matrix has two identical singular +values. By taking the determinants of (14) and (15), the refractive indices can be denoted +by singular values of Jacobian matrices as 𝑛cloak = 1/𝜎J and 𝑛illusion = 1/𝜎Λ. + +Discrete conformal mapping and transformation medium +Review on discrete conformal mapping. It has been demonstrated that an isotropic +refractive index distribution can be achieved by solving equations for equal optical path +length only on rotationally-symmetric surfaces (25). As to the non-rotationally symmetric +cloak, high anisotropy is introduced by the projection mapping that distorts the coordinate +grid (14). However, numerical algorithms for surface parameterization provide possible +conformal mappings for arbitrary surfaces. For example, the angle-based flattening (ABF) +method (45, 46) has been proposed to construct conformal parameterization by +minimizing a punishing functional to decrease angular distortion while its nonlinearity +reduces computational efficiency. Also, the so-called least-squares method (LSCM) (47) +and spectral method (SCP) (48) have been introduced to attain higher efficiency, +benefiting from their linearity. Their disadvantages are free target boundaries and non- +bijectivity, whereas we expect a one-to-one mapping that includes every point on physical +and virtual space with controlled boundaries. Further research, like disk conformal +mapping (DCM) (40), has been reported as a linear and bijective conformal mapping +method but with a fixed disk boundary. Not until boundary first flattening (BFF) (29) +enabled editing boundary as demand were the drawbacks totally eliminated. To deal with +a certain electromagnetic circumstance, one could choose an appropriate algorithm among +the preceding techniques (49, 50). + +Triangulation and Jacobian matrices. Supposing that the conformal mapping reads +𝑓1: ℳ2 → ℳ1 (or 𝑓2: ℳ1 → ℳ2) between manifolds ℳ1 ⊂ ℝ3 and ℳ2 ⊂ ℝ2, as shown in +Fig. S5A, one can find that a simplex 𝒮1 on meshed ℳ1 and its counterpart on meshed ℳ2 +are a pair of similar triangles, which allows 𝒮1 and 𝒮2 to share a same barycentric +coordinate system. This local coordinate system, as shown in Fig. S5B, can represent any + +Manuscript Template +Page 9 of 22 + +point inside the simplex as the linear combination of three vertices and helps quickly +induce the Jacobian matrix of numerical mappings based on triangular mesh +parameterization. For example, the location of the point 𝐪(𝑥′, 𝑦′) on 𝒮2 can be expressed +as 𝑥′ = ∑ +𝜆𝑖𝑥𝑖 +′ +3 +𝑖=1 + and 𝑦′ = ∑ +𝜆𝑖𝑦𝑖 +′ +3 +𝑖=1 + with 𝜆1 + 𝜆2 + 𝜆3 = 1, i.e., a linear combination of +vertices 𝐪1(𝑥1′,𝑦1′), 𝐪2(𝑥2′, 𝑦2′) and 𝐪3(𝑥3′,𝑦3′). For the triangulation mesh, we can +obtain the barycentric coordinates, which read as + +𝜆1 = [(𝑦2 +′ − 𝑦3 +′)(𝑥′ − 𝑥3 +′) + (𝑥3 +′ − 𝑥2 +′)(𝑦′ − 𝑦3 +′)]/det (𝐌), +(17) + +𝜆2 = [(𝑦3 +′ − 𝑦1 +′)(𝑥′ − 𝑥3 +′) + (𝑥1 +′ − 𝑥3 +′)(𝑦′ − 𝑦3 +′)]/det (𝐌), +(18) + +𝜆3 = [(𝑦1 +′ − 𝑦2 +′)(𝑥′ − 𝑥2 +′) + (𝑥2 +′ − 𝑥1 +′)(𝑦′ − 𝑦2 +′)]/det (𝐌), +(19) +where det(𝐌) = |(𝐪1 − 𝐪3) × (𝐪2 − 𝐪3)|, with 𝐪𝑖(𝑥𝑖′, 𝑦𝑖′) being the 𝑖-th vertices (𝑖 = +1, 2, 3). Here, (17), (18) and (19) show that the barycentric coordinate system (𝜆1, 𝜆2, 𝜆3) +can be expressed by the Cartesian coordinate system (𝑥′, 𝑦′). Regarding the point +𝐩(𝑥, 𝑦, 𝑧) on 𝒮1 ⊂ ℝ3, mapped from the point 𝐪 in ℝ2, we have 𝑥 = ∑ +𝜆𝑖𝑥𝑖 +3 +𝑖=1 +, 𝑦 = +∑ +𝜆𝑖𝑦𝑖 +3 +𝑖=1 + and 𝑧 = ∑ +𝜆𝑖𝑧𝑖 +3 +𝑖=1 + as the linear combination of 𝐩1(𝑥1, 𝑦1, 𝑧1), 𝐩2(𝑥2, 𝑦2, 𝑧2) and +𝐩3(𝑥3, 𝑦3, 𝑧3), since 𝒮1 and 𝒮2 share the same barycentric coordinates 𝜆𝑖. As a result, the +Jacobian matrix 𝐉3×2 of the mapping from 𝒮2 ⊂ ℝ2 to 𝒮1 ⊂ ℝ3 can be derived according +to the derivatives of (𝜆1, 𝜆2, 𝜆3) with respect to (𝑥′, 𝑦′), which reads as + +𝐉3×2 = ( +𝜕𝑥′𝑥 +𝜕𝑦′𝑥 +𝜕𝑥′𝑦 +𝜕𝑦′𝑦 +𝜕𝑥′𝑧 +𝜕𝑦′𝑧 +) = +1 +det(𝐌) ( +𝑥1 +𝑥2 +𝑥3 +𝑦1 +𝑦2 +𝑦3 +𝑧1 +𝑧2 +𝑧3 +) ( +𝑦2 +′ − 𝑦3′ +𝑥3 +′ − 𝑥2 +′ +𝑦3 +′ − 𝑦1 +′ +𝑥1 +′ − 𝑥3 +′ +𝑦1 +′ − 𝑦2 +′ +𝑥2 +′ − 𝑥1 +′ +). +(20) +In a similar manner, one can derive the Jacobian matrix 𝚲2×3 of the numerical mapping +from 𝒮1 to 𝒮2; alternatively, one may calculate the Moore–Penrose pseudoinverse of 𝐉3×2 +as 𝚲2×3 (31). By calculating the Jacobian matrices 𝐉3×2 or 𝚲2×3 on each simplex, the +information of mapping 𝑓1 or 𝑓2 can be fully described. + +Simulation methods +FEM simulation. The wave behavior of electromagnetic devices is simulated using the +finite element method. The geometric model is an optical thin-film waveguide whose +thickness is less than one fifth of the wavelength. On the outer surfaces of the waveguide, +the perfect electric conductor (PEC) boundary condition is applied to emulate the +propagation of the surface wave on a two-dimensional manifold. Thus, the propagation of +the plane wave or Gaussian beam is restricted within the optical thin film. To mimic an +open and non-reflecting infinite domain, perfectly matched layers (PMLs) are applied on +the boundary of the propagating plane. The designed medium is configured to the +waveguide as a fitting function interpolated from the discrete data set calculated on extra +dense meshes. + +References +1. J. B. Pendry, D. Schurig, D. R. Smith, Controlling electromagnetic fields. Science. 312, +1780–1782 (2006). +2. U. Leonhardt, Optical conformal mapping. Science. 312, 1777–1780 (2006). +3. H. Chen, C. Chan, Acoustic cloaking in three dimensions using acoustic metamaterials. +Appl Phys Lett. 91, 183518 (2007). + +Manuscript Template +Page 10 of 22 + +4. X. Wang, H. Chen, H. Liu, L. Xu, C. Sheng, S. 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Am Math Soc. 67, 1 (2020). + +Acknowledgments +Funding: This work was supported by the National Natural Science Foundation of China +(NSFC) under grant no. 51977165. + +Author contributions: + +Conceptualization: X.Z., X.M., T.D. + +Manuscript Template +Page 13 of 22 + + +Methodology: X.Z., H.D., X.G., T.D. + +Investigation: X.Z., H.D., X.G. + +Visualization: X.Z., H.D., X.G. + +Supervision: X.M., T.D. + +Writing—original draft: All authors. + +Writing—review & editing: All authors. + +Competing interests: The authors declare that they have no competing interests. + +Data and materials availability: All data needed to evaluate the conclusions in the paper +are present in the paper and/or the Supplementary Materials. Raw data and corresponding +simulation data are available upon request. + +Figures and Tables + +Fig. 1. The conformal mapping between manifolds. (A) A light beam crossing a curved +two-dimensional manifold ℳ embedded in ℝ3. (B) A light beam crossing a flat +two-dimensional manifold ℳ′ in ℝ2. The manifold ℳ is 𝑢𝑣-parameterized and +both manifolds are plotted with coordinate grid. One can obtain the manifold ℳ in +(A) from ℳ′ in (B) through a certain analytic or numerical mapping 𝑓: ℳ′ → ℳ. + + +A +McR3 +u +V +f: M'-M +B +M' c R? +XManuscript Template +Page 14 of 22 + + +Fig. 2. The field and medium distribution for cloaks. Normalized electric field +distribution of surface electromagnetic wave cloaks achieved by (A) anisotropic +relative permeability and (B) isotropic refractive index. (C) Components of +anisotropic relative permeability, 𝜇𝑥𝑥, 𝜇𝑦𝑦 and 𝜇𝑥𝑦, applied in (A) and isotropic +refractive index 𝑛c,double applied in (B). The excitation is a 𝑧-polarized plane wave +with a magnitude of |𝐸𝑧| = 1 V/m; and the wavelength in free-space is 𝜆0 = +20 mm. The bump with a height of 1.25𝜆0 is located in the center of the square +waveguide with a width of 12𝜆0. For the elliptical boundary, the semi-minor and +semi-major axis length are 𝑎 = 3.75𝜆0 and 𝑏 = 5𝜆0, respectively, along with 𝑥- +and 𝑦-axes. + + + +A +12^o +Ez +B +c +μxx +Py +μxy +Nc.double +1 +0.63 +0.63 +1 +0.19 +0.19 +0.83Manuscript Template +Page 15 of 22 + + +Fig. 3. The field and medium distribution for illusions. Normalized electric field +distribution of surface electromagnetic wave scattering. (A) Scattering on the +single-camelback bump when filled with homogeneous medium. (B) Illusion of the +single-camelback bump appearing on the plane. (C) Illusion of the single- +camelback bump appearing on the double-camelback bump. (D) Isotropic +refractive indices: 𝑛i,plane for the elliptic region in (B) and 𝑛i,double for the double- +camelback bump in (C). The elliptical base profiles in (A), (B) and (C) are the +same. + +A +15^o +B +c +D +ni,plane +ni.double +1.25 +0.85 +1.21Manuscript Template +Page 16 of 22 + + + +Fig. 4. The field and medium distribution for devices. Normalized electric field +distribution on surface electromagnetic wave devices. (A) Luneburg lens; (B) +Eaton lens; and (C) Black hole. Gaussian beam is applied to demonstrate their +functions. (D) Isotropic refractive indices; 𝑛Luneburg for Luneburg lens in (A), +decimal logarithm of 𝑛Eaton for Eaton lens in (B), real and imaginary part of +𝑛Blackhole for black hole in (C). + + + + +12Ao +A +focus +[EI2 +B +c +D +nLuneburg +log1o(nEaton) +real(nglaothole) +imag(nglaochole) +1.37 +0 +8 +2.44 +0.083 +0.097Manuscript Template +Page 17 of 22 + +Supplementary Materials for + +• +Controlling Electromagnetic Surface Waves with Conformal +Transformation Optics + +Xiaoyu Zhao et al. + +*Corresponding author. Email: tydong@mail.xjtu.edu.cn. + + + + + +This PDF file includes: + +Figs. S1 to S5 + + + +Manuscript Template +Page 18 of 22 + + +Fig. S1. (A) Normalized electric field distribution of surface electromagnetic wave scattering on +double-camelback bump when filled with homogeneous medium. (B) Quasi-conformal ratio 𝑸 of +the mapping applied to design the cloak shown in Fig. 2A. (C) Curvature index 𝝆. (D) +Wavelength index 𝒘. + + + +A +Ez +B +C +D +p = 0.54 +Q +d +W +1.028 +0 +10.2 +0 +0.24Manuscript Template +Page 19 of 22 + + +Fig. S2. (A) Normalized electric field distribution of the surface electromagnetic wave cloak on +single-camelback bump achieved by (B) isotropic refractive index 𝒏𝐜,𝐬𝐢𝐧𝐠𝐥𝐞. (C) Quasi-conformal +ratio 𝑸 of the mapping applied to design the cloak shown in (A). (D) Curvature index 𝝆. (E) +Wavelength index 𝒘. + + + +A +12^o +B +C +D +E +nc.single +p=0.39 +Q +d +W +0.8 +1.011 +0 +14.5 +0 +0.21Manuscript Template +Page 20 of 22 + + +Fig. S3. A quasi-conformal mapping between two manifolds embedded in ℝ𝟑 constructed by +cascading two mappings between ℝ𝟑 and ℝ𝟐. (A) A single-camelback manifold 𝓜𝟏 embedded in +ℝ𝟑. (B) The plane region 𝓜𝟐 in ℝ𝟐 mapped from 𝓜𝟏 through mapping 𝒇𝟏. (C) The double- +camelback manifold 𝓜𝟑 embedded in ℝ𝟑 mapped from 𝓜𝟐 through mapping 𝒇𝟐. + + + +A +M1 CR3 +fi : M1 → M2 +B +M2 C IR2 +fz : M2 → M3 +c +M3 CR3Manuscript Template +Page 21 of 22 + + +Fig. S4. (A) Normalized electric field 𝑬𝒛, (B) background field 𝑬𝐛𝒛 and (C) scattering field 𝑬𝒛 − +𝑬𝐛𝒛 of the hemisphere surface wave cloak achieved by (D) isotropic refractive index 𝒏𝐜,𝐬𝐩𝐡𝐞𝐫𝐞. (E) +Quasi-conformal ratio 𝑸 of the mapping applied to design the cloak shown in (A). (F) Curvature +index 𝝆. (G) Wavelength index 𝒘. The radius of the hemisphere is 𝟓𝝀𝟎. + + + +A +Ez +1 +15^o +B +Epz +Ez- Ebz +c +0.75 +0.75 +D +E +F +G +p = 1.57 +nc.sphere +W +0.5 +1.012 +0 +35 +0 +0.2Manuscript Template +Page 22 of 22 + + +Fig. S5. (A) Simplices 𝓢𝟏 and 𝓢𝟐 as triangle elements in the mesh of double-camelback manifold +𝓜𝟏 embedded in ℝ𝟑 and the region 𝓜𝟐 in ℝ𝟐, related by quasi-conformal mappings 𝒇𝟏 and 𝒇𝟐. +(B) The same barycentric coordinates on simplices 𝓢𝟏 and 𝓢𝟐. + + +A +B +Pi(X1,Y1,Z1) +Si +MICR3 +P=入1P1+22P2+2P3 +p(x,y,z) +Si +P3(X3,Y3,Z3) +P2(X2,Y2,Z2) +f2:Mi→M2 +f:M2-→M1 +Barycentric +Coordinates +=A,(x,y-x(x.y,z) +q1(xi,yi) +S2 +M2CR2 +q=191+22+3 +q(x,y) +S2 +q3(x3,y3) +q2 (x2,y2) \ No newline at end of file diff --git a/KtAyT4oBgHgl3EQff_jL/content/tmp_files/load_file.txt b/KtAyT4oBgHgl3EQff_jL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ff3f6b207218b360c53209654578aaacda770932 --- /dev/null +++ b/KtAyT4oBgHgl3EQff_jL/content/tmp_files/load_file.txt @@ -0,0 +1,825 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf,len=824 +page_content='Manuscript Template Page 1 of 22 FRONT MATTER Title • Controlling Electromagnetic Surface Waves with Conformal Transformation Optics Authors Xiaoyu Zhao,1† Hong Deng,1† Xiaoke Gao,1† Xikui Ma,1 Tianyu Dong1* 1School of Electrical Engineering, Xi’an Jiaotong University, Xi’an 710049, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' †These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' *To whom correspondence should be addressed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' E-mail: tydong@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='xjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Abstract The application of transformation optics to the development of intriguing electromagnetic devices can produce weakly anisotropic or isotropic media with the assistance of quasi- conformal and/or conformal mapping, as opposed to the strongly anisotropic media produced by general mappings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' however, it is typically limited to two-dimensional applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' By addressing the conformal mapping between two manifolds embedded in three-dimensional space, we demonstrate that electromagnetic surface waves can be controlled without introducing singularity and anisotropy into the device parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Using fruitful surface conformal parameterization methods, a near-perfect conformal mapping between smooth manifolds with arbitrary boundaries can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Illustrations of cloaking and illusions, including surface Luneburg and Eaton lenses and black holes for surface waves, are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Our work brings the manipulation of surface waves at microwave and optical wavelengths one step closer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Teaser Waves can be controlled at will on arbitrary open surfaces without holes, showing fascinating applications such as invisible bumps for surface waves, reproducing scatterings of one bump on other smooth surfaces, and controlling light beams on surfaces to focus, to bend and/or to be absorbed akin to black holes without visible scatterings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' MAIN TEXT Introduction Since its inception in the design of electromagnetic cloaks (1, 2), transformation optics (TO) has proven to be a powerful tool for understanding and customizing the physics in acoustics (3), optics (4), mechanics (5), thermodynamics (6, 7), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Following the groundbreaking work of cloaking, a number of other electromagnetic devices have been reported within the theoretical framework of TO, such as electromagnetic concentrators (8, 9), field rotators (10), optical lenses (11, 12) and optical illusion devices (13, 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' In practice, however, traditional TO often yields significant anisotropy in a designed medium (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Thus, metamaterials are often used to infer spatial changes from coordinate transformation geometry, which is based on the mathematical equivalence between geometry and material (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Manuscript Template Page 2 of 22 To reduce the anisotropy of the functional medium induced by TO, various approaches have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' By constructing mapping in non-Euclidean space, for instance, it is possible to remove singular points formed by traditional TO (17), hence minimizing anisotropy in part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' But for wavelengths comparable to the size of the transform region, non-Euclidean TO may perform even worse (18);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' thus, several research projects focus on conformal or quasi-conformal mappings to achieve isotropy (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=" In ℝ2, the concept of a carpet cloak that resembles a flat ground plane is successfully realized with an isotropic medium produced by minimizing the Modified-Liao functional under sliding boundary conditions (20), or equivalently by constructing the quasi-conformal mapping via solving inverse Laplace's equations (21)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Although the concept of carpet cloak has been extended to ℝ3 by the extrusion or revolution of a two-dimensional refractive index profile to control the reflection of free-space waves, it is only applicable to surfaces with translational or rotational symmetry (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Previous research has focused largely on controlling propagating waves by TO, whereas less attention has been attached to the manipulation of surface waves (12, 23, 24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Perfect surface wave cloaks have been proposed by equating the optical path length of a ray traversing a flat plane with a homogeneous refractive index to the optical path on a curved surface with an angle-dependent refractive index for two orthogonal paths (25, 26), which have been experimentally validated (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Although an electrically large object may be hidden by such a cloak with an inhomogeneous isotropic medium, this approach is limited to rotationally symmetric surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' By linking the governing eikonal equations on a virtual flat plane and on a curved surface by transformation optics, the projection mapping yields surface wave cloaks for non-rotationally symmetric geometries but with high anisotropy (14, 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Considerable effort has been devoted to reducing such anisotropy by employing efficient numerical conformal algorithms such as boundary first flattening (29), yet only non-rotationally symmetric surfaces with circular boundary are investigated (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' In this work, we show how to manipulate surface waves on smooth manifolds embedded in ℝ3 within the framework of conformal TO, requiring an effective isotropic material under the regime of geometrical optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 1 illustrates a conformal surface mapping between two smooth manifolds in ℝ2 and ℝ3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=', 𝑓: ℳ′ → ℳ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The curved manifold ℳ shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 1A has been 𝑢𝑣-parameterized and the mesh grid can be regarded as the mapping result of the Cartesian coordinate system {𝑥′, 𝑦′} in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' When the mapping is conformal or quasi-conformal, the face element d𝑆 remains right-angled, indicating that elements are just scaled up with little distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' From the local coordinate systems on d𝑆 and d𝑆′ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S5), one can derive the Jacobian matrix 𝐉 of mapping 𝑓 with two singular values 𝜎J1 = 𝜎J2 = 𝜎J that state equal scaling in two orthogonal directions (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Consequently, an isotropic cloaking medium distribution 𝑛 = 1/√det(𝐉) = 1/𝜎J may be obtained based on the conformal TO (19), representing the ratio of line element d𝑙′ in virtual space to the scaled element d𝑙 in physical space for compensating optical path length (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' As a result, light propagating on curved ℳ behaves as propagating on flat ℳ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' In practice, it is more convenient to describe mesh vertices in ℝ3 in a Cartesian coordinate system {𝑥, 𝑦, 𝑧} and the Jacobian derived from the local coordinate system forms an asymmetric rank-two matrix 𝐉3×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' In addition, the possible quasi-conformal mappings can be measured by the conformality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=', the ratio 𝑄 = max(𝜎J1/𝜎J2, 𝜎J2/𝜎J1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' A unity ratio 𝑄 allows an effective cloaking medium expressed as 𝑛cloak = 1/√𝜎J1𝜎J2 for every face element (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Manuscript Template Page 3 of 22 Results Having obtained a conformal mapping between the manifolds ℳ ∈ ℝ3 and ℳ′ ∈ ℝ2, we first design an isotropic surface wave cloak under the perspective of conformal TO and compare its performance with the traditional surface wave cloak with anisotropic medium (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Simulations were conducted on a double-camelback bump with an elliptical base profile embedded in ℝ3, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' In comparison with the scattering when the surface has no index profile (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S1A), one can observe that the surface wave cloaking is successfully achieved by two distinct approaches: one induced by the projection mapping proposed in (14) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 2A) and the other originated from the proposed quasi-conformal mapping (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 2B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The corresponding material characteristics for the two types of cloaks are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 2C, indicating that the former is strongly anisotropic while the latter is almost isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' In addition, the isotropic refractive index 𝑛c,double (the subscript "c" denotes the cloak, and "double" denotes the double-camelback bump) ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='83 to 1, which decreases as the bump height rises because a longer geometrical distance need to be compensated by a smaller refractive index in order to attain equal optical path length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The proposed scheme based on conformal TO has achieved near-perfect surface wave cloaking while eliminating the anisotropy in the transformation medium that the traditional scheme presents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The distribution of 𝑛c,double in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 2C outlines an asymmetric geometric profile, manifesting that the effectiveness of this scheme is independent from any symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Such an achievement demands mappings with high conformality rather than those bringing large distortion such as the projection mapping (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The numerical method we adopt here (29) can obtain a quasi-conformal mapping with 𝑄 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='03, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S1B, which is sufficient for designing an effective isotropic cloaking medium distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' As the antithesis of cloaking, optical illusion devices can reproduce the scattering characteristics of a specific object on other objects through a transformation medium (13, 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 3A depicts the surface electromagnetic wave scattered by a single-camelback bump ℳ filled with homogeneous material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Traditionally, if one wants to reproduce its scattering on a plane region ℳ′, the quasi-conformal mapping for designing the illusion device is 𝑓′: ℳ → ℳ′ with a Jacobian matrix 𝚲2×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 3B shows the accurately recurring scattering characteristics on plane region ℳ′ filled with 𝑛i,plane = 1/√𝜎Λ1𝜎Λ2 (the subscript "i" denotes the illusion, and "plane" denotes the plane region), where 𝜎Λ1 and 𝜎Λ2 are singular values of 𝚲2×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 3C illustrates that the double- camelback bump filled with a carefully designed isotropic medium distribution can reproduce the same scattering pattern as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 3A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Such an illusion is realized by cascading two conformal mappings described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=', 𝑓1 from ℝ3 (virtual space) to ℝ2 (intermediate space), and 𝑓2 from ℝ2 to the ℝ3 (physical space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Thus, the illusion medium for the double-camelback bump reads 𝑛i,double = 𝑛i,plane ∙ 𝑛c,double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 3D displays the profiles of 𝑛i,plane (for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 3B) and 𝑛i,double (for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 3C), respectively, which range from 1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='25 (𝑛i,plane) and from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='85 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='21 (𝑛i,double).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The scattering pattern of the single-camelback bump (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 3A) has been successfully reproduced on the plane region (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 3B) and on the double-camelback bump (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 3C), which demonstrates that the proposed scheme is a general solution to illusion design on smooth two-dimensional manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The cascading method to construct mappings between manifolds embedded in ℝ3 can even tackle surfaces with different base profiles, since a Manuscript Template Page 4 of 22 conformal mapping between simply-connected regions in ℝ2 exists according to the Riemann mapping theorem (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Moreover, the quasi-conformal ratios 𝑄 of the two mappings for the double-camelback and single-camelback bump are smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='03 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S1B) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='012 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S2C), respectively, implicating that the cascaded mapping meets the requirement for high conformality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The range of 𝑛i,single (1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='25) is the inverse of that of the cloaking refractive index 𝑛c,single (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='8 to 1) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S2B, because the illusion can be regarded as the inverse design of the cloaking such that the Jacobian matrices of their corresponding mappings are the Moore–Penrose pseudo-inverse of each other (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Now that the wave behavior on the curved manifold can be manipulated flexibly, it is natural to consider designing various complicated devices on it, such as surface wave Luneburg lens, Eaton lens and black hole for surface waves (12, 23, 33, 34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Traditional designs are usually based on spherical or circular profiles with a constant radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' While for an elliptical profile without a constant radius, we adopt the distance from the point on the ellipse to the center, also the coordinate origin, as the generalized radius, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=', 𝑅(𝜃) = √(𝑎 cos 𝜃)2 + (𝑏 sin 𝜃)2 (35–37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Thus, the refractive index of the considered Luneburg lens can be expressed as 𝑛L(𝑟, 𝜃) = √2 − (𝑟/𝑅(𝜃))2, (1) where 𝑟 = √𝑥2 + 𝑦2 and 𝜃 = arctan(𝑦/𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Similar to the traditional circular Luneburg lens, such a distribution retains 𝑛L = 1 on the boundary and 𝑛L = √2 at the center 𝑟 = 0 (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Next, the medium distribution for a Luneburg lens on the double-camelback bump can be expressed as 𝑛Luneburg = 𝑛c,double ⋅ 𝑛L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 4A, two Gaussian beams with a free-space wavelength 𝜆G = 50 mm are incident along the 𝑥-direction at the position ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='8𝑏 on the 𝑦-direction and reflected by the Luneburg lens to interfere at the focus point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The focal distance reads 20𝜆G that is identical to the unit circular Luneburg lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' For the Eaton lens, the refractive index 𝑛E reads as 𝑛E(𝑟, 𝜃) = √2𝑅(𝜃)/𝑟 − 1, (2) which can approach infinity when 𝑟 = 0, leaving a singular point to be cared for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 4B describes that a Gaussian beam going along the 𝑥-direction bends to the inverse 𝑥- direction after passing through the Eaton lens on the double-camelback bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The proposed surface wave Luneburg and Eaton lenses may be deployed in optical imaging, signal acquisition and novel designs for surface wave microwave antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Another functional device that can rotate beam propagation is the peripheral of the two-layer optical black hole, where light is compelled to travel in a spiral path into the absorbing medium at the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The piece-wise refractive index distribution function 𝑛B can be expressed as 𝑛B(𝑟, 𝜃) = { 1, 𝑟 > 𝑅(𝜃) 𝑅(𝜃)/𝑟, 𝑟𝑐 ⋅ 𝑅(𝜃) < 𝑟 < 𝑅(𝜃) 1/𝑟𝑐 + i𝛾, 𝑟 < 𝑟𝑐 ⋅ 𝑅(𝜃) , (3) where 𝑟𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='4 is the scaling factor of the internal ellipse core compared with the base profile and 𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='1 is the loss factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The refractive index distribution 𝑛Blackhole = 𝑛c,double ⋅ 𝑛B on the double-camelback bump is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The real part of material parameters is matched on the inner boundary, and the imaginary part for absorbing energy ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='083 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='097 only exists in the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The same Gaussian Manuscript Template Page 5 of 22 beam that was used for the Eaton lens is employed, and the result in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 4C shows that the beam bends around 90∘ before it reaches the inner boundary and is absorbed by the lossy core without reflection, showing potential application in interference reduction and energy harvesting for electronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Note that, the overall sizes of the simulation models are larger than ten times the operating wavelength, demonstrating that the proposed scheme is capable of managing surface wave behaviors on electrically large objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Moreover, the excellent performance of these functional surface wave devices demonstrates that, based on the proposed scheme, a variety of novel devices may be realized on smooth curved manifolds, which may facilitate the development of miniaturized and integrated photonic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Discussion Our theory and method are based on geometrical optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' It requires small curvature and little variation in wavelength (see (7) and (8) in Materials and Methods), which can be expressed as 𝑤 = |∇𝜆| = |∇(𝜆0/𝑛)| = 𝜆0|∇𝑛|/𝑛2 ≪ 1, (4) 𝜌 = |𝑅𝑖𝑗|𝜆2 = |𝐾𝑔𝑖𝑗|(𝜆0/𝑛)2 = det(𝑔𝑖𝑗) 𝐾2𝜆0 2/𝑛2 = 𝐾2𝜆0 2/𝑛6 ≪ 1, (5) where 𝑅𝑖𝑗 is the Ricci curvature tensor, 𝐾 is the Gaussian curvature, and 𝑔𝑖𝑗 is the metric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Both the wavelength index 𝑤 and the curvature index 𝜌 are inversely proportional to powers of the refractive index 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' In order to prevent 𝑤 and 𝜌 from increasing drastically, a height lower than half of the base radius is favorable, and thereby the optical path length can be compensated with a near-unity refractive index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' On this basis, requirements (4) and (5) demand shorter wavelength 𝜆0 and smoother geometric structure to ease the changing rate |∇𝑛| and the Gaussian curvature 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' As a negative example, a hemisphere surface wave cloak is reviewed and results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S4, whose refractive index 𝑛c,sphere is between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='5 and 1 and the maximum of quasi-conformal ratio 𝑄 is smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The visible scattering appearing in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S4C implies the failure of geometrical optics because of the high curvature index 𝜌 > 20 residing in the right-angle connection between the hemisphere and the plane, as is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S4F, and the average curvature index 𝜌̅ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='57 is also larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The non-smooth connection causes the phase distortion in the backward scattering, and the maximum of the forward scattering |𝐸𝑧 − 𝐸b𝑧|max = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='75 V/m implies a phase difference arcsin (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='75) = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='6∘ resulted from the reconstruction of wave fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' In comparison, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S1C and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S2D display the average curvature index 𝜌̅ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='54 for double-camelback bump and 𝜌̅ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='39 for single-camelback bump, respectively, both satisfying the requirement (5) and leaving near-zero 𝜌 on smooth boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' One may notice that the wavelength index 𝑤 for the cloaks shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S1D, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S2E and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S4E is smaller than unity everywhere because it is related to lower powers of 𝜆0 and 𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' thus, it is much easier to meet the requirement of (4) compared to (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' These selected curvature and wavelength characteristics that validate the approximation of geometrical optics are indispensable for the excellent performance of electromagnetic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The isotropic case that determines the expression of requirements (4) and (5) is based on the conformal or quasi-conformal mappings between two-dimensional manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Benefiting from the rapid development in conformal parameterization, a series of mapping methods can be employed to design surface wave carpet cloak (29, 39, 40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The boundary first flattening (BFF) method (29) adopted in our study can establish near-perfect conformal mappings not only between smooth manifolds but also surfaces with cuspidal Manuscript Template Page 6 of 22 points, such as sharp corners and cone singularities, offering exhilarating promise for wave manipulation on more complicated surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' In addition, there are algorithms aimed at constructing quasi-conformal mappings between high-genus manifolds (41, 42), which can be used to deal with phase regulation on surfaces with holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' One noteworthy idea is to map a high-genus surface to a zero-genus plane region by transforming holes to slits (43, 44) that implies the possibility for the scheme conducted in simply-connected regions to manipulate wave behaviors on multiply-connected surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' By reasonably utilizing advanced algorithms for a variety of particular cases, our method has the potential to be a universal scheme for controlling surface electromagnetic waves on an arbitrary two- dimensional manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' In summary, we have proposed a general method to manipulate electromagnetic waves on smooth two-dimensional manifolds without rotational symmetry by means of a certain isotropic refractive index distribution derived from the quasi-conformal mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The relationship between medium and mappings is induced from the wave equation on the manifold under the geometrical optics approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Numerical quasi-conformal algorithms are introduced to construct mappings between manifolds, and consequent functional mediums are validated by cloaking surfaces and generating illusions on plane regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' By cascading mappings between ℝ2 and ℝ3 to obtain a mapping between ℝ3, we succeed in reproducing the scattering of a surface on another surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' In addition, functional devices such as surface Luneburg lenses, surface Eaton lenses, and black holes for surface waves are designed based on carpet cloaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Finally, the indices required by geometrical optics are reviewed to demonstrate the validity of the approximation on simulation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Our method paves the way for the regulation of surface electromagnetic waves on any two-dimensional manifold, and can be utilized to control surface waves in other fields, such as acoustics, mechanics, and thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Materials and Methods Conformal transformation optics for surface waves Wave equation on curved manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The concept of transformation medium stems from the equivalence between geometry and media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=" Within the Einstein summation convention, the Maxwell's wave equation for the electric field ∇ℳ × ∇ℳ × 𝐄 − 𝜇0𝜀0𝜕𝑡 2𝐄 = 0 in free space can be expressed as (16) 𝛻𝑗𝛻𝑗𝐸𝑖 − 𝑅𝑖𝑗𝐸𝑗 − 𝑐0 −2𝜕𝑡 2𝐸𝑖 = 0, (6) where 𝑐0 = 1/√𝜇0𝜀0 is the light velocity in free space;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 𝑅𝑖𝑗 is the Ricci tensor of the considered geometry ℳ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Supposing that the electromagnetic waves are confined nearby a curved surface ℳ embedded in ℝ3 as surface waves, its local plane wave solution reads as 𝐸𝑖 = ℰi𝑒i𝜑 with constant complex amplitudes ℰi, where the phase reads as 𝜑 = k ⋅ r − 𝜔𝑡 with the wave vector 𝐤 = ∇ℳ𝜑 and angular frequency 𝜔 = −𝜕𝑡𝜑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' For surface waves, the wave vector k lies in the tangent space of the curved surface ℳ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=', 𝐤 ∈ 𝒯(ℳ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Thus, (6) can be simplified and approximated in the regime of geometrical optics where the wavelength 𝜆 = 2𝜋/𝑘 varies slowly with distance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=', |∇ℳ𝜆| ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (7) In addition, the effective curvature of the curved surface should be small enough compared to the wavelength so that the assumption of locally plane waves is valid, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=', |𝑅𝑖𝑗|𝜆2 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (8) Manuscript Template Page 7 of 22 As a result, inserting 𝐸𝑖 = ℰi𝑒i𝜑 into (6) and considering that the (spatial and temporal) derivatives of ℰi vanish, one can obtain the dispersion relation for the surface wave propagating on ℳ, which reads as 𝑘2 = 𝑘𝑗𝑘𝑗 = 𝑔𝑖𝑗𝑘𝑖𝑘𝑗 = 𝜔2/𝑐0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (9) Here, 𝑔𝑖𝑗 is the induced metric tensor for the curved surface ℳ, which can be determined from the transformation Jacobian matrix from the manifold ℳ′ in ℝ2 to ℳ (31),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Wave equation on a flat plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Alternatively, if ℳ is flat (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=', 𝑅𝑖𝑗 = 0) and filled with anisotropic medium denoted by relative permeability tensor 𝜇𝑖𝑗, (6) becomes ∇ × ∇ × 𝐄 − 𝜇0𝜀0𝝁 ⋅ 𝜕𝑡 2𝐄 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (10) Suppose that the electromagnetic waves are confined nearby ℳ and the electric field 𝐄 is perpendicularly polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' In a Cartesian coordinate system, if ℳ can be placed into 𝑥𝑦 plane, we focus on the case that the electric field vector 𝐄 lies in the normal space of the flat plane ℳ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=', 𝐄 ∈ 𝒩(ℳ), and the global wave solution may read as 𝐸𝑧 = ℰ𝑧𝑒i𝜑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Thus, the phase 𝜑 is independent of 𝑧 and the wave vector just lies on the plane as 𝐤 = (𝑘𝑥, 𝑘𝑦, 0), because a flat plane is coincident with its tangent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Since the flat manifold ℳ has a zero-curvature tensor, the condition (7) holds naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Once the other condition (8) that wavelength varies slowly is satisfied, one may disregard the derivatives of complex amplitude after inserting 𝐸𝑧 = ℰ𝑧𝑒i𝜑 into (10) and obtain the dispersion relation for the surface wave propagating on ℳ, which reads as (𝜇𝑥𝑥𝑘𝑥2 + 2𝜇𝑥𝑦𝑘𝑥𝑘𝑦 + 𝜇𝑦𝑦𝑘𝑦2)/det(𝝁) = 𝜔2/𝑐0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' By excluding consideration of the particular polarization, the dispersion equation can be recast within the Einstein summation convention as 1 det(𝝁) 𝜇𝑖𝑗𝑘𝑖𝑘𝑗 = 𝜔2 𝑐0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (11) Transformation medium and geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' For electromagnetic waves that behave identically on two manifolds, one can obtain the equivalence between geometry and material properties by comparing (9) and (11), which yields 𝜇𝑖𝑗 det(𝝁) = 𝑔𝑖𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (12) The relative permeability tensor 𝜇𝑖𝑗 actually creates an illusion on the flat plane because a spatial point filled with medium 𝝁 is equivalent to be with a metric 𝒈 = det(𝝁)𝝁−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' If the local Cartesian coordinate system at this point is aligned along the orthogonal eigenvectors of 𝝁, the real and symmetric permeability tensor will reduce to diag(𝜇𝑥, 𝜇𝑦, 𝜇𝑧) so that the square of the line element on 𝑥 direction is d𝑠2 = 𝑔𝑥𝑥d𝑥2 = 𝜇𝑦𝜇𝑧d𝑥2, which is also the square of optical path length in curved free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' In comparison to d𝑠2 = 𝑛𝑥2d𝑥2 on the flat manifold, one can derive 𝑛𝑥2 = 𝜇𝑦𝜇𝑧 and similar results on 𝑦 and 𝑧 directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Consequently, the relationship between the relative permeability tensor 𝝁 and the refractive index tensor 𝒏 may be expressed as 𝒏2 = det(𝝁)𝝁−1 and one may further obtain 𝒏illustion 2 = 𝒈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (13) by referring to (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Manuscript Template Page 8 of 22 Surface transformation and TO medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The metric tensor in equation (13) is induced from the mapping 𝑓: ℳ′ → ℳ and can be constructed by the Jacobian matrix 𝐉3×2 as 𝒈 = 𝐉T𝐉 (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Nevertheless, we prefer to associate 𝒏illusion with the Jacobian matrix 𝚲2×3 that represents the transformation from ℝ3 (virtual space) to ℝ2 (physical space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Actually, the asymmetric Jacobian matrices 𝐉3×2 and 𝚲2×3 can be denoted as the Moore–Penrose pseudo-inverse of each other (31), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=", 𝐉 = 𝚲†, where the superscript '†' denotes pseudo- inverse." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Thus, one can rewrite the equivalence (13) as 𝒏illustion 2 = 𝒈 = 𝐉T𝐉 = (𝚲𝚲T)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (14) Similar relationship can be obtained for cloaking medium 𝒏cloak and corresponding Jacobian matrix 𝐉3×2 from ℝ2 (virtual space) to ℝ3 (physical space) as 𝒏cloak 2 = (𝐉T𝐉)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (15) For the mapping between ℝ3 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S3), which is formed by cascading two transformations between ℝ3 and ℝ2, the consequent medium for the illusion can be recast as the combination of the cloaking and illusion refractive index tensors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=', 𝒏illustion 2 = (𝚲1𝚲1 T)−1 ⋅ (𝐉2 T𝐉2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (16) where 𝚲1 and 𝐉2 are Jacobian matrices for mappings 𝑓1 and 𝑓2, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' In particular, when the mappings are conformal, the refractive index becomes isotropic, and the corresponding Jacobian matrix has two identical singular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' By taking the determinants of (14) and (15), the refractive indices can be denoted by singular values of Jacobian matrices as 𝑛cloak = 1/𝜎J and 𝑛illusion = 1/𝜎Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Discrete conformal mapping and transformation medium Review on discrete conformal mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' It has been demonstrated that an isotropic refractive index distribution can be achieved by solving equations for equal optical path length only on rotationally-symmetric surfaces (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' As to the non-rotationally symmetric cloak, high anisotropy is introduced by the projection mapping that distorts the coordinate grid (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' However, numerical algorithms for surface parameterization provide possible conformal mappings for arbitrary surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' For example, the angle-based flattening (ABF) method (45, 46) has been proposed to construct conformal parameterization by minimizing a punishing functional to decrease angular distortion while its nonlinearity reduces computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Also, the so-called least-squares method (LSCM) (47) and spectral method (SCP) (48) have been introduced to attain higher efficiency, benefiting from their linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Their disadvantages are free target boundaries and non- bijectivity, whereas we expect a one-to-one mapping that includes every point on physical and virtual space with controlled boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Further research, like disk conformal mapping (DCM) (40), has been reported as a linear and bijective conformal mapping method but with a fixed disk boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Not until boundary first flattening (BFF) (29) enabled editing boundary as demand were the drawbacks totally eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' To deal with a certain electromagnetic circumstance, one could choose an appropriate algorithm among the preceding techniques (49, 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Triangulation and Jacobian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Supposing that the conformal mapping reads 𝑓1: ℳ2 → ℳ1 (or 𝑓2: ℳ1 → ℳ2) between manifolds ℳ1 ⊂ ℝ3 and ℳ2 ⊂ ℝ2, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S5A, one can find that a simplex 𝒮1 on meshed ℳ1 and its counterpart on meshed ℳ2 are a pair of similar triangles, which allows 𝒮1 and 𝒮2 to share a same barycentric coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' This local coordinate system, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S5B, can represent any Manuscript Template Page 9 of 22 point inside the simplex as the linear combination of three vertices and helps quickly induce the Jacobian matrix of numerical mappings based on triangular mesh parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' For example, the location of the point 𝐪(𝑥′, 𝑦′) on 𝒮2 can be expressed as 𝑥′ = ∑ 𝜆𝑖𝑥𝑖 ′ 3 𝑖=1 and 𝑦′ = ∑ 𝜆𝑖𝑦𝑖 ′ 3 𝑖=1 with 𝜆1 + 𝜆2 + 𝜆3 = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=', a linear combination of vertices 𝐪1(𝑥1′,𝑦1′), 𝐪2(𝑥2′, 𝑦2′) and 𝐪3(𝑥3′,𝑦3′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' For the triangulation mesh, we can obtain the barycentric coordinates, which read as 𝜆1 = [(𝑦2 ′ − 𝑦3 ′)(𝑥′ − 𝑥3 ′) + (𝑥3 ′ − 𝑥2 ′)(𝑦′ − 𝑦3 ′)]/det (𝐌), (17) 𝜆2 = [(𝑦3 ′ − 𝑦1 ′)(𝑥′ − 𝑥3 ′) + (𝑥1 ′ − 𝑥3 ′)(𝑦′ − 𝑦3 ′)]/det (𝐌), (18) 𝜆3 = [(𝑦1 ′ − 𝑦2 ′)(𝑥′ − 𝑥2 ′) + (𝑥2 ′ − 𝑥1 ′)(𝑦′ − 𝑦2 ′)]/det (𝐌), (19) where det(𝐌) = |(𝐪1 − 𝐪3) × (𝐪2 − 𝐪3)|, with 𝐪𝑖(𝑥𝑖′, 𝑦𝑖′) being the 𝑖-th vertices (𝑖 = 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Here, (17), (18) and (19) show that the barycentric coordinate system (𝜆1, 𝜆2, 𝜆3) can be expressed by the Cartesian coordinate system (𝑥′, 𝑦′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Regarding the point 𝐩(𝑥, 𝑦, 𝑧) on 𝒮1 ⊂ ℝ3, mapped from the point 𝐪 in ℝ2, we have 𝑥 = ∑ 𝜆𝑖𝑥𝑖 3 𝑖=1 , 𝑦 = ∑ 𝜆𝑖𝑦𝑖 3 𝑖=1 and 𝑧 = ∑ 𝜆𝑖𝑧𝑖 3 𝑖=1 as the linear combination of 𝐩1(𝑥1, 𝑦1, 𝑧1), 𝐩2(𝑥2, 𝑦2, 𝑧2) and 𝐩3(𝑥3, 𝑦3, 𝑧3), since 𝒮1 and 𝒮2 share the same barycentric coordinates 𝜆𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' As a result, the Jacobian matrix 𝐉3×2 of the mapping from 𝒮2 ⊂ ℝ2 to 𝒮1 ⊂ ℝ3 can be derived according to the derivatives of (𝜆1, 𝜆2, 𝜆3) with respect to (𝑥′, 𝑦′), which reads as 𝐉3×2 = ( 𝜕𝑥′𝑥 𝜕𝑦′𝑥 𝜕𝑥′𝑦 𝜕𝑦′𝑦 𝜕𝑥′𝑧 𝜕𝑦′𝑧 ) = 1 det(𝐌) ( 𝑥1 𝑥2 𝑥3 𝑦1 𝑦2 𝑦3 𝑧1 𝑧2 𝑧3 ) ( 𝑦2 ′ − 𝑦3′ 𝑥3 ′ − 𝑥2 ′ 𝑦3 ′ − 𝑦1 ′ 𝑥1 ′ − 𝑥3 ′ 𝑦1 ′ − 𝑦2 ′ 𝑥2 ′ − 𝑥1 ′ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (20) In a similar manner, one can derive the Jacobian matrix 𝚲2×3 of the numerical mapping from 𝒮1 to 𝒮2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' alternatively, one may calculate the Moore–Penrose pseudoinverse of 𝐉3×2 as 𝚲2×3 (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' By calculating the Jacobian matrices 𝐉3×2 or 𝚲2×3 on each simplex, the information of mapping 𝑓1 or 𝑓2 can be fully described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Simulation methods FEM simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The wave behavior of electromagnetic devices is simulated using the finite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The geometric model is an optical thin-film waveguide whose thickness is less than one fifth of the wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' On the outer surfaces of the waveguide, the perfect electric conductor (PEC) boundary condition is applied to emulate the propagation of the surface wave on a two-dimensional manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Thus, the propagation of the plane wave or Gaussian beam is restricted within the optical thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' To mimic an open and non-reflecting infinite domain, perfectly matched layers (PMLs) are applied on the boundary of the propagating plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The designed medium is configured to the waveguide as a fitting function interpolated from the discrete data set calculated on extra dense meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' References 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Lévy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Petitjean, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Ray, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Maillot, Least squares conformal maps for automatic texture atlas generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' ACM Trans Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 21, 362–371 (2002).' metadata={'source': 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Comp Graph Forum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 27 (2008), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='1467-8659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='01289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Peters, Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=', Natick, MA, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Gu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Luo, S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Writing—original draft: All authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Writing—review & editing: All authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Competing interests: The authors declare that they have no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Raw data and corresponding simulation data are available upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Figures and Tables Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The conformal mapping between manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (A) A light beam crossing a curved two-dimensional manifold ℳ embedded in ℝ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (B) A light beam crossing a flat two-dimensional manifold ℳ′ in ℝ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The manifold ℳ is 𝑢𝑣-parameterized and both manifolds are plotted with coordinate grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' One can obtain the manifold ℳ in (A) from ℳ′ in (B) through a certain analytic or numerical mapping 𝑓: ℳ′ → ℳ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=" A McR3 u V f: M' M B M' c R?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' XManuscript Template Page 14 of 22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The field and medium distribution for cloaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Normalized electric field distribution of surface electromagnetic wave cloaks achieved by (A) anisotropic relative permeability and (B) isotropic refractive index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (C) Components of anisotropic relative permeability, 𝜇𝑥𝑥, 𝜇𝑦𝑦 and 𝜇𝑥𝑦, applied in (A) and isotropic refractive index 𝑛c,double applied in (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The excitation is a 𝑧-polarized plane wave with a magnitude of |𝐸𝑧| = 1 V/m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' and the wavelength in free-space is 𝜆0 = 20 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The bump with a height of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='25𝜆0 is located in the center of the square waveguide with a width of 12𝜆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' For the elliptical boundary, the semi-minor and semi-major axis length are 𝑎 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='75𝜆0 and 𝑏 = 5𝜆0, respectively, along with 𝑥- and 𝑦-axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' A 12^o Ez B c μxx Py μxy Nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='double 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='63 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='83Manuscript Template Page 15 of 22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The field and medium distribution for illusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Normalized electric field distribution of surface electromagnetic wave scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (A) Scattering on the single-camelback bump when filled with homogeneous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (B) Illusion of the single-camelback bump appearing on the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (C) Illusion of the single- camelback bump appearing on the double-camelback bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (D) Isotropic refractive indices: 𝑛i,plane for the elliptic region in (B) and 𝑛i,double for the double- camelback bump in (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The elliptical base profiles in (A), (B) and (C) are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' A 15^o B c D ni,plane ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='double 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='21Manuscript Template Page 16 of 22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The field and medium distribution for devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Normalized electric field distribution on surface electromagnetic wave devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (A) Luneburg lens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (B) Eaton lens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' and (C) Black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Gaussian beam is applied to demonstrate their functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (D) Isotropic refractive indices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 𝑛Luneburg for Luneburg lens in (A), decimal logarithm of 𝑛Eaton for Eaton lens in (B), real and imaginary part of 𝑛Blackhole for black hole in (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 12Ao A focus [EI2 B c D nLuneburg log1o(nEaton) real(nglaothole) imag(nglaochole) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='37 0 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='097Manuscript Template Page 17 of 22 Supplementary Materials for Controlling Electromagnetic Surface Waves with Conformal Transformation Optics Xiaoyu Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' Email: tydong@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='xjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' This PDF file includes: Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S1 to S5 Manuscript Template Page 18 of 22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (A) Normalized electric field distribution of surface electromagnetic wave scattering on double-camelback bump when filled with homogeneous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (B) Quasi-conformal ratio 𝑸 of the mapping applied to design the cloak shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (C) Curvature index 𝝆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (D) Wavelength index 𝒘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' A Ez B C D p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='54 Q d W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='028 0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='24Manuscript Template Page 19 of 22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (A) Normalized electric field distribution of the surface electromagnetic wave cloak on single-camelback bump achieved by (B) isotropic refractive index 𝒏𝐜,𝐬𝐢𝐧𝐠𝐥𝐞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (C) Quasi-conformal ratio 𝑸 of the mapping applied to design the cloak shown in (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (D) Curvature index 𝝆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (E) Wavelength index 𝒘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' A 12^o B C D E nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='single p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='39 Q d W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='011 0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='21Manuscript Template Page 20 of 22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' A quasi-conformal mapping between two manifolds embedded in ℝ𝟑 constructed by cascading two mappings between ℝ𝟑 and ℝ𝟐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (A) A single-camelback manifold 𝓜𝟏 embedded in ℝ𝟑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (B) The plane region 𝓜𝟐 in ℝ𝟐 mapped from 𝓜𝟏 through mapping 𝒇𝟏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (C) The double- camelback manifold 𝓜𝟑 embedded in ℝ𝟑 mapped from 𝓜𝟐 through mapping 𝒇𝟐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' A M1 CR3 fi : M1 → M2 B M2 C IR2 fz : M2 → M3 c M3 CR3Manuscript Template Page 21 of 22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (A) Normalized electric field 𝑬𝒛, (B) background field 𝑬𝐛𝒛 and (C) scattering field 𝑬𝒛 − 𝑬𝐛𝒛 of the hemisphere surface wave cloak achieved by (D) isotropic refractive index 𝒏𝐜,𝐬𝐩𝐡𝐞𝐫𝐞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (E) Quasi-conformal ratio 𝑸 of the mapping applied to design the cloak shown in (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (F) Curvature index 𝝆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (G) Wavelength index 𝒘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' The radius of the hemisphere is 𝟓𝝀𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' A Ez 1 15^o 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+page_content='2Manuscript Template Page 22 of 22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (A) Simplices 𝓢𝟏 and 𝓢𝟐 as triangle elements in the mesh of double-camelback manifold 𝓜𝟏 embedded in ℝ𝟑 and the region 𝓜𝟐 in ℝ𝟐, related by quasi-conformal mappings 𝒇𝟏 and 𝒇𝟐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' (B) The same barycentric coordinates on simplices 𝓢𝟏 and 𝓢𝟐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content=' A B Pi(X1,Y1,Z1) Si MICR3 P=入1P1+22P2+2P3 p(x,y,z) Si P3(X3,Y3,Z3) P2(X2,Y2,Z2) f2:Mi→M2 f:M2 →M1 Barycentric Coordinates =A,(x,y x(x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} +page_content='y,z) q1(xi,yi) S2 M2CR2 q=191+22+3 q(x,y) S2 q3(x3,y3) q2 (x2,y2)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAyT4oBgHgl3EQff_jL/content/2301.00352v1.pdf'} diff --git a/M9FOT4oBgHgl3EQf1zQJ/content/tmp_files/2301.12940v1.pdf.txt b/M9FOT4oBgHgl3EQf1zQJ/content/tmp_files/2301.12940v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..071134847810ef5fed6dd7e1c33f2523502d966c --- /dev/null +++ b/M9FOT4oBgHgl3EQf1zQJ/content/tmp_files/2301.12940v1.pdf.txt @@ -0,0 +1,3249 @@ +Krylov methods for large-scale dynamical systems: Application in +fluid dynamics +Ricardo S. Frantz1, Jean-Christophe Loiseau1, and Jean-Christophe Robinet1 +1Arts et M´etiers Institute of Technology, CNAM, DynFluid, HESAM Universit´e, F-75013 +Paris, France +January 31, 2023 +Abstract +In fluid dynamics, predicting and characterizing bifurcations, from the onset of unsteadi- +ness to the transition to turbulence, is of critical importance for both academic and industrial +applications. Different tools from dynamical systems theory can be used for this purpose. In +this review, we present a concise theoretical and numerical framework focusing on practical +aspects of the computation and stability analyses of steady and time-periodic solutions, with +emphasis on high-dimensional systems such as those arising from the spatial discretization of +the Navier-Stokes equations. Using a matrix-free approach based on Krylov methods, we ex- +tend the capabilities of the open-source high-performance spectral element-based time-stepper +Nek5000. The numerical methods discussed are implemented in nekStab, an open-source and +user-friendly add-on toolbox dedicated to the study of stability properties of flows in complex +three-dimensional geometries. The performance and accuracy of the methods are illustrated +and examined using standard benchmarks from the fluid mechanics literature. Thanks to its +flexibility and domain-agnostic nature, the methodology presented in this work can be applied to +develop similar toolboxes for other solvers, most importantly outside the field of fluid mechanics. +1 +Introduction +The transition to turbulence is a long-standing problem in fluid dynamics, pioneered by Osborne +Reynolds [217]. As mathematical tools have developed, a dynamical systems point of view has led +to a better understanding of this phenomenon. Before the advent of computers, theoretical analyses +had to rely on simplifying assumptions, the most important ones being the parallel flow assumption +and that of infinitesimal perturbations forming what we now know today as local stability theory. +For simple shear flows, these assumptions lead to the famous Orr-Sommerfeld-Squire equations +ïÅ ∂ +∂t +ã +∇2 − U′′ ∂ +∂x − 1 +Re∇4 +ò +v = 0, +ï ∂ +∂t + U ∂ +∂x − 1 +Re∇2 +ò +η = −U′ ∂v +∂z , +(1) +where U(y) is the base flow velocity profile, v(x, y, z, t) is the wall-normal velocity component of +the perturbation, and η(x, y, z, t) its wall-normal vorticity component. Despite their simplicity, +these assumptions led to important theorems in hydrodynamic stability theory: Rayleigh’s in- +flection point criterion [214], Fjørtoft [101] or Squire’s theorem [243]. They also led to a better +1 +arXiv:2301.12940v1 [physics.flu-dyn] 30 Jan 2023 + +understanding of non-normality and to the development of nonmodal stability analysis [215, 231]. +Although U(y) is formally a steady solution of the Navier-Stokes equations, a tremendous amount +of understanding has been gained by replacing it with simple approximations such as the Bachelor +vortex sheet model or the piecewise linear approximation of the Blasius boundary layer profile in +the seminal work of Tollmien and Schlichting [229, 256]. Using a normal mode ansatz, the velocity +fluctuation can be decomposed using a Fourier expansion +v(x, y, t) = ˆv(y) exp (i(αx − ωt)) , +(2) +and similarly for the vorticity fluctuation. Determining the stability of the system then amounts to +solving a generalized eigenvalue problem. Depending on the assumptions about the wavenumber α +and the frequency ω, these stability analyses fall into two categories +• Temporal stability: defined by α ∈ R and ω ∈ C. In this context, one aims to determine +whether a fluctuation grows over time at a given streamwise position. +• Spatial stability: defined by α ∈ C and ω ∈ R. In this context, one investigates whether +forcing at a particular frequency causes the perturbation to grow in space while being advected +by the flow. +An important milestone was achieved by Huerre & Monkewitz [127, 128] by letting both α and ω be +complex numbers. Depending on subtle properties of the dispersion relation in the complex plane, +they introduced the dichotomy between absolute and convective instabilities, thus establishing the +first connection between the local stability properties of the flow and its spatiotemporal evolution. +An instability is classified as absolute if the perturbation grows in place (i.e. its group velocity +is zero), otherwise as convective if it grows as it propagates before leaving the region of interest. +For more details, see [73, 127]. This connection between local and global properties of flow was +refined by Monkewitz, Huerre and Chomaz [185] using the WKBJ formalism and led to the weakly +non-parallel flow assumption which is, however, still very limiting and hardly applicable to flow +configurations of practical interest where separation is important. For a good numerical overview +of local stability theory, interested readers are referred to the book by Schmid & Henningson [232]. +This leap in understanding the nature of instabilities allowed for better explanations and ex- +panded the limiting dichotomy of open and closed flows, as the dynamics themselves could now +be categorized as noise amplifiers or oscillators (more details given in [73, 128]). This distinction +is of great importance, for example, for the selection and design of flow control strategies or the +placement of sensors and actuators (see the review by Schmid and Sipp [233] and also Cossu [74]). +Flow configurations of a noise-amplifying nature are much more difficult to control and predict +because the dynamics are sensitive to both the amplitude and the spectral content of the incoming +disturbance (viz. vibrations, acoustics, or turbulence intensity). In such flows, the incoming dis- +turbances can excite otherwise stable modes and begin to extract energy from the base flow while +they are transported downstream. Natural oscillators, on the other hand, are characterized by +the presence of a dominant unstable structure (i.e. a global instability), which locates the physical +mechanism that extracts energy from the base flow in space. When the instability is suppressed, +the flow becomes stable and returns to the laminar state. More complex flow configurations (such +as the jet in crossflow) may exhibit convective noise amplifier behavior or self-sustaining oscillatory +behavior, depending on the combination of control parameters [178]. +Problems such as the one presented in equation (1) can be analyzed theoretically, but in practice, +it is common to discretize the resulting equations in the wall-normal direction using spectral meth- +ods such as Chebyshev polynomials. A famous example is the work of Orszag [195] on the temporal +2 + +stability of the canonical plane Poiseuille flow. At about the same time as [65, 127, 128, 185], com- +puters and numerical methods began to reach sufficient maturity that the (weakly non-)parallel flow +assumption could be relaxed and the spectral decomposition of the Navier-Stokes operator linearized +in the vicinity of a truly two-dimensional base flow started to be computed. In fact, the computation +of 2D baseflows in hydrodynamics began in the 1970s with direct methods [29, 182, 234, 275] before +other, more efficient numerical approaches became popular and years later enabled the computation +of 3D baseflows with increasing computer power. +In the mid-1980s, Jackson [131] and Zebib [278] obtained 2D steady states and computed stabil- +ity analysis on the flow past a circular cylinder using full-matrices (Jackson used iterative methods +to approximate the leading eigenvalues). +At the same time, the first use of a time-dependent +numerical solver and Krylov methods for fluid dynamics seems to point to the work of Erikson +& Rizzi [93], which coincides with the eigenvalue calculations of Tuckerman [262] and Marcus & +Tuckerman [171, 172] on the flow between concentric rotating spheres. Similar numerical methods +have been explored by Goldhirsch et al. [113], Christodoulou & Scriven [66], Tuckerman [260], and +Edwards et al. [89]. +In these works, the prohibitive memory requirements of a matrix-forming +approach are replaced by methods that use more accessible computational resources and are now +commonly referred to as the matrix-free/time-stepping approach. These advances in algorithms +eventually enabled the computation of 3D eigenmodes evolving on 2D solutions in the 1990s. This +began with the work of Natarajan & Acrivos [188] and Ramanan & Homsy [212] on the lid-driven +cavity flow and later the analysis of Barkley & Tuckerman [22] on the perturbed plane Couette +flow. +To distinguish these analyses, which solve the linearized 2D equations, from a local stability +framework relying on a (weakly-) parallel flow approximation, Theofilis et al. [250] has called them +(bi-)global stability analyses. +Since then, the linear stability of numerous two-dimensional flow +configurations have been studied: the lid-driven cavity flow [2, 251], the backward-facing step [146], +or the two-dimensional flow past a bump [90, 91], to name just a few. Because of its importance for +the development of local stability theory, the two-dimensional boundary layer flow has also been +the focus of many investigations [5, 12]. +Although matrix-free methods were already available, it is important to mention that several of +the previously mentioned papers still considered the explicit construction of the linearized Navier- +Stokes operator in combination with standard algebraic solvers to compute its eigenpairs (other +examples include [106, 118, 251]). +For a comprehensive review of research contributions up to +the early 2010s, with particular emphasis on this matrix-forming approach, see Theofilis [252] and +Juniper et al. [132]. +Yet, over the past decade, the time-stepping framework has become increasingly popular and +enabled the investigation of the stability properties of fully three-dimensional flows. A large body of +works has focused on two configurations, namely the jet in crossflow [13, 129, 203] or the boundary +layer flow past three-dimensional roughness elements [43, 46, 48, 67, 145, 154, 162, 276]. +The +stability of lid-driven and shear-driven three-dimensional cavities with spanwise end-walls has also +been investigated in [99, 107, 143, 151, 155, 205]. The same methodology has also been employed +to compute the leading optimal perturbation [231] in magnetohydrodynamic flows [280], or best +exemplified by [35, 36] on backward facing steps and stenotic pipe flows. It has also been used to +solve high-dimensional Ricatti equations for linear optimal control in [235], or to study the stability +properties of flow governed by the compressible Navier-Stokes equations with or without shocks [94, +95, 226]. These include modal and non-modal stability of compressible boundary layers [49, 50, +117, 125, 218], cavities [41, 247, 253, 277], wavepackets in jets [26, 192, 236], transonic buffet [75– +77, 199, 200, 255], including the flow past the NASA Common Research wing Model [254, 255], +wakes [181] and bluff bodies [163, 164, 226, 227]. +3 + +The matrix-free approach has been the key to efficient computation and (Floquet) stability +analysis of time-periodic solutions, beginning with Schatz, Barkley & Swinney [228], followed by +the secondary instability of the flow past a circular cylinder with the canonical work of Barkley & +Henderson [19] and later by the study of Blackburn, Marques & Lopez [34]. Other work includes +the backward-facing step flow [24], the study of the stability properties of a pulsatile stenotic pipe +flow [36], the flip-flop instability in the wake of two side-by-side cylinders [59], the secondary bi- +furcation in a shear-driven cavity flow [28], or the study of the vortex pairing mechanism in a +harmonically forced axisymmetric jet [238]. In parallel with these developments in the hydrody- +namic stability community, similar numerical methods and tools have been explored in the com- +munity looking at turbulence from the point of view of a dynamical system. Matrix-free methods +and clever exploration of flow symmetries can significantly reduce computational costs and allow +the calculation of exact coherent states1 in the turbulent basin of attraction. These exact coher- +ent states include relative periodic orbits [141, 213], chaotic saddles [136, 201, 270, 274] or edge +states [56, 130, 241]. Interested readers may refer to the specialized codes Channelflow [111] or +Openpipeflow [273] and the corpus of related works. It is also noteworthy that in the special case of +weakly non-parallel shear flows, the development of parabolized stability equations [123, 149] paved +the way for the estimation of the neutral stability curves of the Blasius boundary layer [31, 121, 152] +and its secondary instabilities [122, 124], which may involve curvature effects inducing centripetal +G¨ortler vortices [166, 216]. +Building on the framework of Orszag and aiming at the geometric flexibility of the finite ele- +ment method, Patera [202] and Maday & Patera [165] laid the foundation of the spectral element +method (SEM). SEM has since become a popular discretization strategy in computational fluid dy- +namics (CFD). Within the incompressible hydrodynamic stability community, the spectral element +solver Nek5000 [100] has established itself as one of the leading high-performance open-source CFD +codes. Most of the aforementioned three-dimensional stability analyses heavily relied on Nek5000. +Except for the KTH Framework2, relatively few toolboxes have been developed for Nek5000 de- +spite its large user base. Even then, the capabilities of this toolbox (as far as linear stability is +concerned) are limited to simple fixed-point calculations using the selective frequency damping ap- +proach [1], while the leading eigenpairs of the linearized Navier-Stokes operator are calculated using +PARPACK [148, 176], at the expense of introducing new dependencies for the code. Linear stability +analysis capabilities are also available to Nek5000’s brethren, including Nektar++ [58, 135], which +can handle a variety of mesh types, and Semtex [33] designed for spanwise periodic flows, with +examples given by [3, 32, 92, 170, 239]. The finite element code FreeFem++[120] can also be used +to extract the Jacobian matrix directly with examples including [68, 69, 133, 173, 174, 277]. The +solver capabilities can be extended using the StabFem [95] MATLAB suite for compressible flows. +Finally, the FEniCS [6] project is also another consolidated alternative with a Python interface. Re- +cently several other spectral-based Python projects for solving partial differential equations became +available. These include SpectralDNS [186, 187], FluidSim [10, 184], Dedalus [51] and Coral [183]. +The finite volume framework BROADCAST [209] has become available for the study of 2D curvilinear +structured grids with support for nonlinear and linear calculations of compressible flows. +The aim of the present work is by no means to be an exhaustive review, but rather a compre- +hensive introduction to the Krylov methods underlying the most recent works on the stability of +1Exact coherent states can be fixed points in the original reference frame or in a co-moving reference frame (in +which case they are called traveling waves) or true periodic states [82, 261]. +2Freely available at https://github.com/KTH-Nek5000/KTH_Framework. Apart from some linear stability capabil- +ities, this toolbox also provides additional capabilities such as full restarts, turbulence statistics, additional boundary +conditions, or incorporating user variables and various volume forces. +4 + +very large-scale dynamical systems such as fully three-dimensional flows. For that purpose, this +manuscript is organized as follows: first, brief overviews of the theoretical framework and numerical +methods are given in section 2 and section 3, respectively. A collection of examples illustrating the +use of these techniques in fluid dynamics is provided in section 4, while some particular theoretical +and practical points are discussed in section 5. Finally, conclusions and perspectives are given in +section 6. +2 +Theoretical framework +We focus on the analysis of the stability properties of high-dimensional nonlinear dynamical systems, +typically arising from the discretization of partial differential equations such as the incompressible +Navier-Stokes equations. Once discretized, the governing equations are generally expressed as a +nonlinear system of first-order ordinary differential equations +dXj +dt += Fj ({Xi; i = 1, · · · , n} , t) , +where n is the dimension of the discretized system. +Using the notation X and F for the sets +{Xj; j = 1, · · · , n} and {Fj; j = 1, · · · , n}, this system can be written as +dX +dt = F(X, t), +(3) +where X ∈ Rn is the state vector of the system and t is a continuous variable that denotes time. If +the system is supplemented with constraints (e.g. the divergence-free constraint in incompressible +fluid dynamics), F is then understood as the restriction of dynamics in the feasible set of such +constraints. One can also consider the equivalent discrete-time system +Xk+1 = Φτ(Xk), +(4) +where Φτ(X) is the forward map defined as +Φτ(X) = +� τ +0 +F(X(t), t) dt. +(5) +Such a discrete-time system may result from the time discretization of the governing equations +(with τ the sampling period) or in the study of periodic orbits (with τ the period of the orbit). In +practice (and due to discretization errors), an approximation to the action of an operator is made +by computing many (i.e. τ/∆t) small time-steps (i.e. ∆t ≪ 1), each consisting of a rational or +polynomial approximation to the operator. The term timestepper, coined in 2000 by Tuckerman & +Barkley [264], refers to the adaptation of a time integration code to perform bifurcation analyses. +In what follows, we consider an exact operator notation as a shorthand notation for the timestepper +approximation of operators (linear or nonlinear). +In the following sections, we present a definition of fixed points, periodic orbits, and linear +stability analyses. These are the fundamental concepts required to characterize the properties of +the system under investigation. In particular, we focus on the modal (i.e. asymptotic) and non- +modal (i.e. finite-time) stability analysis, which are the classical and a more modern approach that +has become increasingly popular in fluid dynamics in recent decades. +We note that part of the community has also shifted its attention to nonlinear optimal pertur- +bations, which is beyond the scope of this work. Interested readers are referred to the work of [139] +and the references therein. +5 + +2.1 +Fixed points and periodic orbits +Nonlinear dynamical systems, such as equation (3) can admit different solutions or attractors that +form the backbone of their phase space: fixed points (steady dynamics), periodic orbits (periodic +dynamics), tori (quasi-periodic dynamics) or strange attractors (chaotic dynamics). Hereafter, our +attention will be solely focused on fixed points and periodic orbits. +2.1.1 +Fixed points +For a continuous-time dynamical system described by equation (3), the fixed points X∗ are partic- +ular equilibrium solutions satisfying +F(X∗) = 0. +(6) +Similarly, for a discrete-time system described by equation (4), fixed points are solutions to +Φτ(X∗) = X∗ +∀τ. +(7) +These particular solutions are thus characterized by the absence of dynamics: the system is in a +steady state. Since we are dealing with nonlinear equations, both equation (3) and equation (4) +can have a multitude of fixed points. This is illustrated by a dynamical system as simple as the +Duffing oscillator +� +� +� +˙x = y, +˙y = −1 +2y + x − x3. +(8) +Because of the stabilizing cubic term in the y-equation, the Duffing oscillator admits three fixed +points +• a saddle at the origin X∗ = (0, 0), +• two linearly stable spirals located at X∗ = (±1, 0). +These different fixed points, along with typical trajectories, are depicted in figure 1. A multiplicity of +fixed points also occurs in nonlinear dynamical systems as complex as the Navier-Stokes equations, +see for instance [116]. The decision of which of these fixed points is the relevant one from a physical +point of view depends on the problem and is left to the judgment of the user. The computation +of equilibria is a cornerstone for all analyses described in this work. Numerical methods to solve +equation (6) and equation (7) are discussed further in section 3. +2.1.2 +Periodic orbits +The second type of equilibria of interest to us is periodic orbits. Such solutions are characterized +by dynamics repeating themselves after a given period τ ∗, i.e. +X(t + τ ∗) = X(t). +(9) +They can also be understood as fixed points of the forward map Φτ for τ = τ ∗, i.e. +X∗ = Φτ(X∗) +for τ = τ ∗. +(10) +In practice, τ ∗ is often unknown, and therefore one must solve simultaneously for a point X∗ on +the orbit and the period τ ∗. Moreover, any point on the orbit satisfies the equation above so that +equation (10) admits an infinite number of solutions. To close the system, a phase condition often +6 + +Figure 1: Phase portrait of the unforced Duffing oscillator. The red dots denote the three fixed +points admitted by the system. The blue (resp. orange) thick line depicts the stable (resp. unstable) +manifold of the saddle point located at the origin. Grey lines highlight a few trajectories exhibited +for different initial conditions. +needs to be included. A canonical example of a periodic orbit in fluid dynamics is the periodic +vortex shedding in the wake of a two-dimensional cylinder at a low Reynolds number. As for fixed +points, a nonlinear dynamical system may admit multiple periodic orbits, each with its own period +τ ∗. This is particularly true when the system evolves on a strange attractor (chaotic dynamics) on +which an infinite number of unstable periodic orbits (UPO) coexist with arbitrary periods. These +findings go back to Kawahara & Kida [136], who pioneered the extraction of periodic orbits in a +fully three-dimensional plane Couette flow using a Newton root-finding technique. The discovery of +a large number of UPOs buried in turbulent attractors for various flow configurations supports the +view that turbulence is a very high-dimensional dynamical system whose trajectories repeatedly +visit unstable exact coherent structures [87, 137]. Since the discovery of recurrent spatiotemporal +structures concealed in a turbulent attractor, UPOs have proven to be favorable kernels [83] for +predicting turbulence statistics due to their harmonic temporal structure [78]. +These invariant +solutions can be naturally sustained by the flow and have been found to be energetically relevant +for predicting or even reconstructing turbulence statistics (if enough of them are found) [62, 159]. +Despite growing interest since the first UPOs were obtained, the methods used to obtain them +involve brute force, although attempts are being made to change this [197]. +Considering the R¨ossler system [219] +� +� +� +� +� +˙x = −y − z, +˙y = x + ay, +˙z = b + z(x − c), +(11) +for a = 0.1, b = 0.1, and c = 14, figure 2 shows its famous strange attractor as well as two UPOs +embedded in this attractor. These were obtained using a simple shooting Newton method and +continuation initialized with their stable counterparts at lower values of the control parameter c. +7 + +2 +O +-2 +-2 +-1 +0 +1 +2 +XFigure 2: Strange attractor for the R¨ossler system with parameters a = 0.1, b = 0.1 and c = 14. +Colored lines depict a τ1 (red) and τ2 (blue) unstable periodic orbit. +2.2 +Modal and non-modal linear stability +To avoid repetition, we restrict ourselves in this section to the modal and non-modal stability +of fixed points, although these concepts can easily be applied to periodic orbits by replacing the +Jacobian matrix with the monodromy matrix. The linearly stable or unstable nature of the fixed +point X∗ is characterized by the fate of infinitesimal perturbations. If perturbations eventually +decay, the equilibrium is considered stable; otherwise, it is considered unstable. Note, however, +that an infinite time horizon is allowed for the return to equilibrium. Thus, a fixed point can be +classified as stable even if a small perturbation transiently departs very far from it before returning +toward it only asymptotically with t → ∞. This distinction between asymptotic and finite-time +evolution gives rise to the concepts of modal and non-modal stability. The reader will find a detailed +discussion of these concepts in fluid dynamics in [230, 231]. +The dynamics of a perturbation x = X − X∗ is governed by +˙x = F (X∗ + x) . +(12) +Assuming x is infinitesimally small, F(X) can be approximated by its first-order Taylor expansion +around X = X∗, leading to +˙x = Lx, +(13) +where +L = ∂F +∂X, +(14) +is the n × n Jacobian of F. Starting from an initial condition x0, the perturbation at time τ is +given by +x(τ) = exp(τL)x0. +(15) +8 + +UPO1 +UPO2 +Z +xThe operator Mτ = exp(τL) is the exponential propagator of the linearized system and corresponds +to the Jacobian of the forward map Φτ linearized in the vicinity of the fixed point X∗. For periodic +orbits, this linearized propagator is defined as +Mτ = +� τ +0 +L(t) dt, +(16) +with L(t + τ) = L(t) and is known as the monodromy matrix. We again use the notation of the +exact operator as shorthand for its discrete counterpart obtained by integration over many small +timesteps. System operators can be linearized exactly by using a timestepper code to solve the +set of linearized equations derived analytically, or by using automatic differentiation tools. Such +an exact linear operator is in itself a discrete approximation, again requiring temporal integration +over many small timesteps. Alternatively, the action of the linearized operator can be emulated +using less accurate finite-difference approximation [26, 177]. While the computational cost remains +the same for a first-order approximation, it increases with the order of the finite-difference scheme +considered. +2.2.1 +Modal stability +Introducing the (Euclidean) 2-norm of x(τ) and the eigendecomposition +L = VΛV−1, +(17) +one can easily show +exp(2τλr) ≤ ∥ exp(τL)∥2 +2 ≤ κ(V) exp(2τλr), +(18) +where λr = Re(λ1) is the real part of the leading eigenvalue (i.e. with greatest real part) of L +and κ(V) = ∥V∥2∥V−1∥2 is the condition number of the matrix of eigenvectors V (with κ ≥ 1). +Asymptotic stability is characterized by +lim +τ→∞ ∥ exp(τL)x0∥2 = 0 +∀ x0, +(19) +so a sufficient condition is that all eigenvalues of L have a negative real part (equivalent to all +eigenvalues of Mτ being inside the unit circle). The perturbation decaying the slowest is given by +the eigenvector v1 associated with the least stable eigenvalue λ1. Hereafter, a fixed point X∗ is +classified as follows +• if Re(λ1) > 0, the dynamics of simulations initialized with an initial condition non-orthogonal +to the leading eigenvector (i.e. x0 ̸⊥ v1) will grow exponentially rapidly. The fixed point X∗ +is deemed linearly unstable. +• if Re(λ1) < 0, the dynamics of simulations initialized with any initial condition will eventually +decay exponentially fast. The fixed point X∗ is thus linearly stable. +The case Re(λ1) = 0 is special and corresponds to a non-hyperbolic fixed point. Its stability cannot +be determined by the eigenvalues of L alone and one has to resort to weakly nonlinear analysis or +center manifold reduction. Interested readers can refer to [60, 167, 240, 271] for more details. +9 + +2.2.2 +Non-modal stability +The upper bound in equation (18) involves the condition number κ(V) of the matrix of eigenvectors. +This leads to the classification of linear systems into two distinct sets with fundamentally different +finite-time stability properties. Systems for which κ(V) = 1 are called normal operators. In this +case, the eigenvectors of L form an orthonormal set such that +V −1 = VH, +(20) +where superscript H denotes the Hermitian, i.e. complex-conjugate transpose operation. The finite- +time and asymptotic stability properties of the system are identical, and the dynamics cannot +exhibit transient growth: analyzing the spectrum of L is sufficient to fully characterize the system. +When κ(V) > 1, the matrix L is said to be non-normal. This can be defined by introducing the +adjoint operator L† satisfying +⟨y|Lx⟩ = ⟨L†y|x⟩, +(21) +where ⟨·|·⟩ is a suitable inner product (typically the inner product induced by the ℓ2 norm), along +with appropriate boundary conditions. Non-normality then corresponds to the fact that L and its +adjoint L† do not commute, i.e. +L†L ̸= LL†. +(22) +Its eigenvectors no longer form an orthonormal basis for Rn and the dynamics may exhibit transient +growth. +From a physical point of view, transient growth can be understood as a constructive +interference involving almost colinear eigenvectors. The larger the non-normality of L, the larger +the maximum transient growth with perturbations being (possibly) amplified by several orders +of magnitude before the exponential decay eventually takes over (assuming all the eigenvalues +have negative real parts). In the most extreme scenario, this non-normality is characterized by L +admitting a non-diagonalizable Jordan block leading to algebraic growth. More details on adjoint +operators are provided in [126, 160, 257]. +Given this observation, one can now ask a more subtle question about the stability of the fixed +point X∗, namely +How far from the fixed point X∗ can an arbitrary perturbation x0 go (or equivalently to +what extent can it be amplified) at a finite time τ? +The answer to this question can be obtained by solving the following optimization problem +G(τ) = max +x0 +∥ exp(τL)x0∥2 +2 +∥x0∥2 +2 +, +(23) +where ∥ exp(τL)∥2 +2 is the vector-induced matrix norm optimizing over all possible initial conditions +x0 and G(τ) is the maximal amplification gain of the perturbation at time τ. Introducing the +singular value decomposition of the exponential propagator Mτ +Mτ = UΣVH, +(24) +the maximum gain is simply given by +G(τ) = σ2 +1, +(25) +where σ1 is the largest singular value of Mτ. The optimal initial condition x0 is then given by +the first right singular vector (i.e. x0 = v1) while the associated response is x(τ) = σ1u1, where +u1 is the leading left singular vector. When/if one has access to the adjoint operator, computing +10 + +these quantities can also be recast as an eigenvalue problem (EVP) rather than a singular value +decomposition (SVD). +In fluid dynamics, this concept of non-normality and optimal perturbations leads to a better +understanding of the formation and ubiquity of velocity streaks in the transition to turbulence of +wall-bounded shear flows [38–40, 258]. It also sheds some light on the importance of shear layer +instability [25, 35, 57]. Extension to periodic orbits has been considered for instance in [36, 170]. +Although not considered herein, a similar concept exists in the frequency domain, leading to a +resolvent analysis (see [231] for more details). +Illustration +Let us illustrate the concepts of optimal perturbation and transient growth using a simple flow +configuration. For this purpose, consider the incompressible flow of a Newtonian fluid induced by +two flat plates moving in opposite directions in the plane, as sketched in figure 3(a). The resulting +flow, known as plane Couette flow, is given by +U(y) = y. +It is a linearly stable fixed point of the Navier-Stokes equations. Despite its linear stability, subcrit- +ical transition to turbulence due to finite amplitude perturbations can occur at Reynolds numbers +as low as Re = 325 [168]. Without delving too much into the mathematical and physical details +of such a subcritical transition, part of the explanation can be given by linear optimal pertur- +bation analysis. The dynamics of an infinitesimal perturbation x = �v +η�T , characterized by a +certain wavenumber k = αex + βez, is governed by the Orr-Sommerfeld-Squire equations (matrix +counterpart of equation (1)) written as +d +dt +ïv +η +ò += +ïLOS +0 +C +LS +ò ïv +η +ò +, +(26) +with v the wall-normal velocity component of the perturbation and η its wall-normal vorticity. LOS +denotes the Orr-Sommerfeld operator, while LS and C represent the Squire operator and the cou- +pling term, respectively. For certain pairs of wavenumbers, this Orr-Sommerfeld-Squire operator is +highly non-normal. In particular, perturbations with non-zero spanwise wavenumbers can experi- +ence large transient growth. This is illustrated in figure 3(b), where the evolution of the optimal +gain G(τ) is shown for different spanwise wavenumbers. The maximum amplification over all target +times and wavenumber pairs is Gopt ≃ 100. The initial perturbation x0 is shown in figure 3(c). +This perturbation corresponds to streamwise-oriented vortices that eventually lead to streamwise +velocity streaks, as shown in figure 3(d). Although this perturbation eventually decays exponen- +tially fast in a purely linear framework, it has been shown that its transient amplification even at +moderately small amplitude can be sufficient to trigger the transition to turbulence when used as +an initial condition in a nonlinear simulation [160]. For more details on subcritical transitions and +the extension of optimal perturbation analysis to nonlinear operators, interested readers may refer +to [231] and [139]. +2.3 +Bifurcation analysis +The eigenvalue analysis of the Jacobian matrix L or monodromy matrix Mτ plays a key role +in determining the type of bifurcation that occurs when a control parameter (e.g. the Reynolds +number) is varied. In the remainder of this section, a brief overview of the standard bifurcations +and their correspondence to eigenvalues is given for completeness (see figure 4 for a schematic +representation). +11 + +Figure 3: Illustration of the optimal perturbation analysis for the plane Couette flow at Re = +300 for streamwise wavenumber α = 0: (a) Schematic of the flow, (b) Optimal gain curve for +different spanwise wavenumbers β, (c) Optimal perturbation in-plane velocity (v, w) and (d) optimal +response out of plane velocity (u) for β = 2. The optimal perturbation consists of streamwise +oriented vortices, while the corresponding response at time T consists of high and low-speed streaks. +Reproduced from [156]. +Figure 4: Eigenvalue patterns associated with the standard bifurcations encountered for fixed points +(a and b), and for limit cycles (c to e). In each case, the shaded region indicates the stable part +of the spectrum (i.e. the lower complex half-plane for fixed-point stability, and the unit disk for +periodic orbits). +12 + +100 +β=2 +β=3 +h +50 +β= 4 +0 +50 +100 +150 +200 +250 +T +b +-1 +0 +1 +2 +-2 +0 +2 +Z +Z +d(a) Pitchfork/Saddle-node +(b) Hopf +说[入] +况[入] +[] +(c) Pitchfork/Saddle-node +(d) Period-doubling/Flip (e) Neimark-Sacker/Secondary-Hopf +s[μ] +sμ] +[n] +[n]2.3.1 +Bifurcations of fixed points +The asymptotic stability properties of fixed points are related to the eigenvalues of the linearized +operator L. Bifurcation analysis is concerned with how these properties evolve as the parameters +of the system are varied. For simplicity, we will only consider situations where a single parameter is +varied. The value of the control parameter at which a change in stability occurs is a bifurcation point. +The bifurcations most commonly encountered in mechanics are the pitchfork, saddle-node, and Hopf +bifurcations. Each leads to a qualitatively different behavior before and after the bifurcation point. +Pitchfork and Hopf bifurcations come in two flavors, namely subcritical and supercritical (depending +on whether the solutions created at the bifurcation point are themselves stable or not). +Pitchfork bifurcation +This type of bifurcation is often encountered in systems with symmetries. +The canonical example in mechanics is that of a flexible beam on top of which a static load is applied. +Below a critical load, the beam remains upright. As the load increases, the beam suddenly buckles +to the left or right. The upright position becomes linearly unstable, and two new stable equilibria +are created as the bifurcation point is crossed. Mathematically, the pitchfork bifurcation can be +distilled into the following normal form +˙x = λx ± x3, +(27) +with the sign in front of the cubic term determining whether the bifurcation is super(-) or sub(+) +critical. From a linear stability point of view, the linearized operator has a purely real eigenvalue +going from being stable (i.e. λ < 0) to unstable (i.e. λ > 0) as we cross the bifurcation point. This +is a necessary, albeit insufficient, condition to conclude that the bifurcation is a pitchfork Moreover, +nonlinear analyses (or simulations) are required in order to conclude whether it is supercritical or +subcritical. Examples from fluid dynamics include the flow in a sudden expansion channel [86, 146], +the flow past a sphere [94, 188, 227], the three-dimensional cavity flow [2, 151, 205] or the Rayleigh- +B´enard convection between two infinite plates or inside an annular loop [157]. +Hopf bifurcation +The second type of bifurcation commonly encountered is the Andronov-Poincar´e- +Hopf bifurcation (or simply Hopf bifurcation). Below the bifurcation point, the system admits a +single fixed point and asymptotically stationary dynamics. As the bifurcation point is crossed, the +fixed point changes stability, and a limit cycle associated with periodic dynamics is created in its +vicinity. The Hopf bifurcation can be distilled into the following normal form [82], written as +® ˙r = σr ± r3, +˙ϕ = ω + αr2, +(28) +where r is the amplitude of the oscillations, ϕ their phase, and ω the frequency of the oscillation. +The linearized system has a complex-conjugate pair of eigenvalues that change from stable (i.e. +σ < 0) to unstable (i.e. σ > 0) when crossing the bifurcation point. The imaginary part (ω) of this +complex-conjugate pair of eigenvalues then dictates the oscillation frequency. Nonlinear analysis +(or simulation) is required to determine whether it is super- or subcritical. Examples from fluid +dynamics include the broad class of so-called flow oscillators such as the two-dimensional cylinder +flow [11, 18, 19, 108, 169, 180, 193, 194, 240, 272, 278, 279], the lid-driven and shear-driven cavity +flows [17, 53, 179, 220, 240, 277], the jet in crossflow [13, 63, 129] or the roughness-induced boundary +layer flow [46, 67, 154]. +13 + +2.3.2 +Bifurcations of periodic orbits +The linearization of the time-periodic flow map Φτ(X) in the vicinity of the periodic orbit X∗ leads +to the monodromy matrix (sometimes also known as the time-shift operator). As for fixed points, +their eigenvalues (known as characteristic or Floquet multipliers) dictate the asymptotic stability of +the periodic orbit under consideration. The stability problem describes the development of small- +amplitude perturbations during one period of evolution. If all Floquet multipliers lie inside the +unit disk, the orbit is characterized as asymptotically linearly stable, otherwise as unstable. +Bifurcations occur when one of these Floquet multipliers (or a complex-conjugate pair) steps +outside the unit circle when the control parameter is varied. Physically, the moduli of such Floquet +multipliers express the orbit rate of expansion (or contraction) per unit of time (i.e. per period +of oscillation). +As an aside, in cases where the limit cycle occurs as an autonomous nonlinear +oscillation (not forced), the set of Floquet multipliers contains a unit mode tangent to the limit +cycle which corresponds to the time derivative of the base flow. A limit cycle in which at least one +Floquet multiplier is greater than one expands and is therefore called an unstable periodic orbit +(UPO). +The most common bifurcations in this context are the pitchfork bifurcation, the period-doubling +bifurcation (also known as a flip bifurcation) and the Neimark-Sacker bifurcation (see figure 4 for a +schematic). Once again, these are associated with qualitatively different evolutions of the dynamics +below and above the bifurcation point. +Pitchfork bifurcation +As for its fixed point counterpart, the pitchfork bifurcation of periodic +orbits is most often encountered in systems with spatial symmetries and comes in two flavors (super- +and subcritical). A canonical example of such pitchfork bifurcations in fluid dynamics is the three- +dimensionalization of the periodic vortex shedding in the wake of a circular cylinder [19, 23]. Below +the critical Reynolds number Rec ≃ 189, the flow is strictly two-dimensional and exhibits the well- +known time-periodic von K`arm`an vortex street. All Floquet multipliers lie within the unit circle. +When the Reynolds number is increased, a Floquet multiplier leaves the unit circle at µ = 1 and +a pitchfork bifurcation occurs3. Given the synchronous nature of the pitchfork bifurcation, the +frequency of the vortex shedding remains unchanged, but the spatial structure of the vortices is no +longer spanwise invariant: the flow becomes three-dimensional. If one denotes by An the amplitude +of this three-dimensionalization after n periods, the corresponding normal form is given by +An+1 = µAn ± |An|2An, +(29) +where µ is the associated Floquet multiplier and the sign of the cubic term determines whether the +bifurcation is super- or subcritical. Note that, as for fixed points, other types of bifurcations (e.g. +saddle-node) are associated with a Floquet multiplier exiting the unit circle at µ = 1. +Period-doubling bifurcation +The second type of bifurcation commonly encountered is the +period-doubling bifurcations. +They are also known as flip or subharmonic bifurcations. +In this +situation, a Floquet multiplier exits the unit circle at µ = −1 as the bifurcation point is crossed. +In the supercritical case, the periodic orbit that was stable below the bifurcation point becomes +unstable, and a new orbit with twice the period takes its place. Considering a discrete-time system, +3Because of the spanwise-invariance of the unstable periodic orbit, the dominant Floquet multiplier (i.e. with the +largest magnitude) is actually a double eigenvalue with one eigenvector exhibiting a sine dependence in the spanwise- +direction while the other exhibits a cosine dependence. In this case, the bifurcation is formally known as a circle +pitchfork bifurcation. +14 + +the most famous example of this period-doubling bifurcation is the logistic map +xn+1 = µxn(1 − xn). +(30) +In a continuous-time framework, this bifurcation can be found in the famous R¨ossler system. Fig- +ure 2 shows two such orbits. The one in blue, denoted as UPO1 with a period τ1 ≃ 6, loses its +stability via a period-doubling bifurcation at c = 5.376 ± 0.001. Above this critical value, the peri- +odic orbit (denoted as UPO2 in figure 2) is created with a period τ2 ≃ 12. This second orbit then +loses its stability through another period-doubling bifurcation (at a critical value c = 7.771±0.001) +and a new orbit with twice the period (i.e. τ3 ≃ 24) is created. Systems presenting period-doubling +bifurcations often exhibit a subharmonic cascade to chaos, a universal behavior of dynamical sys- +tems put forth by Feigenbaum and others in the late 1970s. In the context of fluid dynamics, such a +subharmonic cascade was shown to occur in a confined Rayleigh-B´enard cell as the Rayleigh number +is increased in the seminal work of Libchaber et al. [150]. It was also observed by Buzug et al. [52] +experimentally in Taylor-Couette flow and numerically in Couette flow [140]. In plane Couette flow, +a first bifurcation leads to the formation of a spatially evolving periodic orbit (also called relative +periodic orbits or traveling waves) at Re = 236.1 [161]. Above Re ≈ 240.40, the system undergoes +a period-doubling cascade. At Re = 240.46 this cascade leads to a chaotic attractor with expo- +nentially diverging trajectories that sporadically visit the various previously created UPOs. More +recently, this subharmonic cascade has also been observed numerically in rotating plane Couette +flow under certain conditions [79]. Period-doubling bifurcations are also observed in harmonically +forced shear layers and jets where they give rise to vortex pairing, see [238] for an example. +Neimark-Sacker bifurcation +The last type of bifurcation we will consider is the Neimark- +Sacker bifurcation named after the works of Neimark [190] in 1959 and Sacker [224] in 1964 (for an +overview of their work see the book by Arnold [8]). This bifurcation is the equivalent of the Hopf +bifurcation of a fixed point for periodic orbits and is therefore sometimes called a secondary Hopf +bifurcation. From the point of view of linear stability, it is associated with a complex conjugate pair +of Floquet multipliers leaving the unit disk at an angle that is neither 0 nor π, according to µ = e±iω. +If the new frequency is rationally related to that of the periodic orbit, the dynamics immediately +after the bifurcation remains periodic, but with a different period. Rationally related frequencies +always form a set of measure zero in the set of possible imaginary parts of the Floquet exponent. +If the new frequency is irrationally related to that of the periodic orbit, the dynamics becomes +quasiperiodic immediately after the bifurcation. In this case, the corresponding phase space object +changes from a simple periodic orbit (below the bifurcation point) to a torus (above the bifurcation +point) [53, see fig. 13] in which the fundamental frequency and the smaller new frequency map +the large and small circles of the torus, respectively. The dynamics following a secondary Hopf +bifurcation can be much more complicated, exhibiting phenomena such as frequency locking, high- +order synchronization (Arnold’s tongues), and devil’s staircase, which are beyond the scope of this +review. Interested readers are referred to the book by Pikovsky, Rosenblum & Kurths [207]. +To help reveal the structure of the attractor, one can plot the intersections of the trajectories +with a plane, called a Poincar´e section, which intersects the attractor. As the trajectories cross the +plane in different positions, a Poincar´e section on a torus-shaped object will continuously cover a +circle [30] (provided the temporal signal is long enough). The Fourier spectrum of a periodic system +that becomes quasiperiodic is characterized by the appearance of a new peak at a frequency f2 that +is incommensurate with the fundamental frequency f1. Due to nonlinear interactions between these +two frequencies, other peaks may appear in the spectra, which can be easily explained by the linear +combinations |n1f1 ±n2f2|, where n1 and n2 are integers. In fluid dynamics, such a bifurcation has +15 + +been shown to occur in the wake of two side-by-side cylinders [59], with the corresponding instability +known as the flip-flop instability, as well as in a two-dimensional shear-driven cavity [53, 147]. +Beyond quasiperiodic dynamics +Over the years, the transition from periodic and quasi- +periodic dynamics to chaos has been studied in detail [114, 248]. When the control parameter +is varied, steady, periodic, and quasi-periodic systems can undergo a progressive loss of stability +and the appearance of chaos by following characteristic paths known as routes to chaos [7], with +the most relevant ones being: +• The Feigenbaum path: The system undergoes a cascade of period-doubling bifurcations; +• The Ruelle-Takens-Newhouse route: gradual generation of incommensurable frequencies by a +sequence of (Hopf) bifurcations; +• The Pomeau-Manneville scenario: the existence of an intermittent alternation of regular +phases and chaotic bursts. +These characteristic routes are reviewed in Eckmann [88] and can be identified by time-series +analysis of physical or numerical experiments based on classical Fourier analysis to more com- +plex phase-portrait reconstructions [42], delayed embedding [115, 249] or recurrence analysis [175] +techniques. +In 1971, Ruelle and Takens [221] showed that when the nonlinearity (or coupling) of a quasi- +periodic system increases, Hopf bifurcations lead to an increase in the dimension of the torus, +eventually becoming structurally unstable and collapses into a strange attractor with non-integer +fractal dimension. The route known as Ruelle-Takens-Newhouse (RTN) [191, 221] is associated +with the direct appearance of chaos after the formation of a T 3 torus (or even T 4 torus in some +cases). In perhaps even rarer cases, the T 3 torus can undergo a rapid devil’s staircase to chaos +[134]. Flows undergoing the RTN path include the convergent-divergent channel flow [119], the +Taylor-Couette flow [72] or the flow in the highly curved toroidal pipe flow [55], or even higher +dimensional tori in T n with n ≥ 3 [114, 150, 196]. +The path of intermittency introduced by Pomeau and Manneville [208] in the 1980s involves an +alternation between periodic and chaotic dynamics, although all system parameters remain constant +and free of noise (Manneville [167]’s book gives a clear overview). Beyond the bifurcation point, +dynamical systems with intermittency exhibit bursts of irregular motion with higher amplitude +amidst regular motion with lower amplitude. The duration of the irregular bursts increases with +the control parameters until the chaotic dynamics predominate. Depending on the value of the +Floquet multiplier at the bifurcation point, there are different evolutionary trajectories for the +decay of the periods of laminar phases. In the theory of intermittent transitions, Floquet stability +analysis provides three classifications for intermittency: Type I is associated with a pitchfork +bifurcation (the Floquet multiplier crosses the unit circle at +1 at the bifurcation point), type +II with a Neimark-Sacker bifurcation (two complex conjugate eigenvalues), and type III with a +period-doubling bifurcation (the Floquet multiplier crosses the unit circle at -1 at the bifurcation +point). Note that the latter two cases require a subcritical character of the bifurcation for the +appearance of intermittency [30]. +3 +Numerical methods +Numerous tools exist to study low-dimensional dynamical systems such as the R¨ossler or Lorenz [158] +systems. These include AUTO [84, 85] in Fortran or pde2path [266, 267] and MatCont [81] in MAT- +16 + +LAB. PyDSTool [71] offers similar capabilities in Python, while BifurcationKit.jl [269] is a corre- +sponding Julia package. Except for BifurcationKit.jl, most of them rely on standard numerical +linear algebra techniques that do not scale well for very high-dimensional problems. Moreover, they +do not necessarily integrate easily with parallel programming, which is ubiquitous when simulating +discretized partial differential equations such as Navier-Stokes. Thus, extra care may be needed to +interface with the particular data structure of the original code (see Algorithm 1). +This section provides a brief overview of the standard iterative techniques used to compute fixed +points or periodic orbits of very high-dimensional dynamical systems and to study their stability +properties. These techniques rely on Krylov subspaces and associated Krylov decompositions [244] +introduced in section 3.1. The Newton-Krylov method for fixed-point computations and its exten- +sion to periodic orbits are discussed in section 3.2. Finally, the use of Krylov techniques to compute +the leading eigenvalues or singular values of the linearized operator to characterize its modal and +non-modal stability properties are discussed in section 3.3. In what follows, we will assume that a +time-stepping simulation code is available to simulate the nonlinear system, i.e. the time-stepping +code returning Xk+1 = Φτ(Xk). Similarly, we will assume that a linearized version of this code can +be used to calculate the matrix-vector product Mτx by time-marching the equations, where the +operator Mτ is either the (numerically approximated) exponential propagator for fixed points or the +monodromy matrix for periodic orbits. For non-modal stability analysis, we furthermore assume +that an equivalent time-stepping code is available to approximate the matrix-vector product M† +τx +where M† +τ is the exponential propagator or monodromy matrix built using the corresponding ad- +joint linear operator L†. The methods advocated here and in [82, 156] can all be easily implemented +with very few modifications into existing codes. +3.1 +Krylov subspaces and the Arnoldi factorization +In [245], the American mathematician Gilbert W. Stewart listed six of the most important ma- +trix decompositions. These include the pivoted LU decomposition, the QR decomposition, the +spectral (i.e. eigenvalue) decomposition, the Schur decomposition, and the singular value decom- +position (SVD). The introduction of each of these decompositions into numerical linear algebra +has revolutionized matrix computations. Nevertheless, a seventh approximate factorization should +be included in this list, namely, the Arnoldi factorization. Introduced by Walter Edwin Arnoldi +in 1951 [9] it relies on the concept of Krylov subspaces [142], named after the Russian applied +mathematician Alexei Krylov. Today, these subspaces are the workhorses of large-scale numerical +linear algebra and form the foundations of numerous iterative linear solvers such as the minimal +residual method (MINRES) [198] or the generalized minimal residual method (GMRES) [222]. The +book Iterative methods for sparse linear systems by Y. Saad [223] is probably the most complete +reference for such techniques. Given a matrix A ∈ Rn×n and a starting vector x ∈ Rn, a Krylov +subspace of dimensions m can be constructed by repeated applications of A, leading to +Km(A, x) = �x, Ax, A2x, · · · , Am−1x� . +(31) +Introducing the matrix +K = �x +Ax +A2x +· · · +Am−1x� , +(32) +the above sequence can be recast as the following Krylov factorization +AK = KC + eT +mr, +(33) +17 + +where C ∈ Rm×m is a companion matrix of the form +C = +� +������ +0 +0 +0 +· · · +0 +c1 +1 +0 +0 +· · · +0 +c2 +0 +1 +0 +· · · +0 +c3 +... +... +... +... +... +... +0 +0 +· · · +· · · +1 +cm +� +������ +. +(34) +The coefficients ci are computed based on a least-squares procedure such that the residual r is not +in the span of the previous m Krylov vectors, i.e. rT K = 0, and ∥r∥2 is minimized. If ∥r∥ = 0, then +the columns of K span an invariant subspace, and the eigenvalues of C are a subset of those of A. +If the starting vector x is random and m is large enough, K most likely tends towards the invariant +subspace of A associated with the eigenvalues having the largest magnitudes. Note however that, +as m increases, the last Krylov vectors become increasingly collinear by virtue of the applied power +iteration and, consequently, the matrix KT K is increasingly ill-conditioned. This companion-based +Krylov factorization is thus of little use in practice due to its numerical instability given finite +arithmetic. +A simple remedy to this numerical instability is to iteratively construct each new Krylov vector +such that it is orthonormal to all previously generated vectors instead of simply applying the +power iteration. Starting from a vector x1 (with ∥x1∥2 = 1), the Krylov basis can be iteratively +constructed by the following algorithm. +Algorithm 1 – Arnoldi factorization +Given : the n × n matrix A, and a unit-norm vector v1. +1. w = Av1 ; α1 = v∗ +1w ; +r = w − v1α ; V1 = [v1] ; H1 = [α]. +2. For j = 1, 2, · · · , m − 1 +(a) Add the residual from the previous iteration into the Krylov basis +βj = ∥r∥2 ; vj+1 = β−1 +j r ; +Vj+1 = �Vj +vj+1 +� ; ‹ +Hj = +ï Hj +βje∗ +j +ò +; +(b) Compute the residual associated with this new Krylov basis +w = Avj+1 ; h = V∗ +j+1w ; +r = w − Vj+1h ; +(c) Update the upper Hessenberg matrix accordingly +Hj+1 = +î‹ +Hj +h +ó +. +After m steps, this leads to the Krylov factorization +AV = VH + eT +mr, +(35) +known as the Arnoldi factorization. In this factorization, the matrix V ∈ Rn×m is orthonormal +(i.e. VT V = I) and H ∈ Rm×m is an upper Hessenberg matrix (almost triangular matrix with +18 + +zero entries below the first subdiagonal). Once again, the residual vector r is the component of +the (m + 1)th Krylov vector not in the span of V (i.e. rT V = 0). Knowledge of the n × m matrix +V and the m × m matrix H can then be used to approximate the dominant (i.e. with the largest +absolute value) eigenvalues and eigenvectors of A or to obtain a reasonable solution to +Ax = b, +at a reduced computational cost compared to direct inversion using Gaussian elimination or LU +techniques. The Arnoldi factorization is at the heart of the widely used GMRES technique for +solving large linear systems presented in Algorithm 3. Other Krylov factorizations exist such as +the Lanczos factorization for Hermitian matrices (where H reduces to a tridiagonal matrix) or +the Krylov-Schur factorization introduced by Stewart [244] (where H is in Schur form), enabling +simple restarting strategies for computing the dominant eigenvalues and eigenvectors of A when +the available RAM is a limiting factor. +3.2 +Newton-Krylov method for fixed points and periodic orbits +For low-dimensional dynamical systems, fixed point (resp. periodic orbit) computations can easily +be performed using the standard Newton method already implemented in numerous languages (e.g. +scipy.optimize.fsolve in Python). These implementations (see Algorithm 2) are often quite +generic and rely on direct solvers for the inversion of the Jacobian matrix L (resp. monodromy +matrix Mτ). Due to the sheer size of the linear systems resulting from the discretization of par- +tial differential equations, this approach however does not scale favorably. Coupling these generic +solvers with an existing time-stepping code may moreover require extra layers of code because of +the particular data structure used in the simulation and possibly its parallel computing capabilities. +Although libraries such as PETSc [14–16] or Trilinos [259] exist, these may also require extra devel- +opment to interface with an existing well-established code. They moreover add extra dependencies +which might complicate the deployment of the resulting applications on a large set of computing +platforms with different operating systems (e.g. from laptops for development to supercomputing +facilities for production runs). +With the goal of extending the capabilities of an existing time-stepping code with a minimum +number of modifications and dependencies, section 3.2.1 (resp. +section 3.2.2) presents a time- +stepping formulation of the Newton-Krylov algorithm to compute fixed points (resp. +periodic +orbits). A similar algorithm has already been introduced by [138] and [82] and in ChannelFlow [111, +112]. As stated previously, we will assume only that we have the nonlinear time-stepper returning +Xk+1 = Φτ(Xk), +(36) +and its linear counterpart +xk+1 = Mτxk, +(37) +where Mτ is either the exponential propagator or the monodromy matrix. We will also assume +that a routine to compute the m-step Arnoldi factorization (see section 3.1) has been implemented. +Along with the inclusion of a direct eigensolver (such as links to LAPACK), this implementation +is the only major development needed to extend the capabilities of an existing time-stepping code. +Benefits from this implementation outperform its development costs as it paves the way for a GM- +RES implementation leveraging all the utilities of the existing time-stepping code. Once available, +this m-step Arnoldi factorization routine can also be readily used to compute the leading eigenval- +ues and eigenvectors of the high-dimensional linearized operator with no extra development (see +section 3.3). +19 + +3.2.1 +Fixed point computation +In a time-stepper formulation, fixed points are solutions to +X = Φτ(X), +(38) +for arbitrary integration time τ. Alternatively, they are the roots of +F(X) = Φτ(X) − X. +(39) +In fluid dynamics, numerous approaches have been proposed in the literature to compute these +fixed points while circumventing the need to implement a dedicated Newton solver into an existing +time-stepping code. +For example, one can cite the selective frequency damping method (SFD) +proposed by ˚Akervik et al. [1] and its variants, or BoostConv [70]. Although they require relatively +minor modifications to an existing simulation code, they suffer from a number of limitations, such +as slow convergence, which was explored in [156]. Because it relies on a temporal low-pass filter, +the selective frequency damping procedure is, moreover, unable to compute saddle nodes, i.e. fixed +points having at least one unstable eigendirection associated with a purely real eigenvalue. +Assuming the m-step Arnoldi factorization has already been implemented, it requires only a +relatively modest effort to integrate it into a dedicated GMRES solver. In doing so, the roots +of equation (39) can be easily computed using the Newton-Krylov technique. The Jacobian of +equation (39) is given by +J = exp(τL) − I, +(40) +where L is the linearized operator around the current estimate X. The matrix-vector product Jx +thus requires calling the linearized time-stepper (i.e. to compute exp(τL)x), with x ∈ Rn being the +Newton correction. +Algorithm 2 – Newton-Krylov solver +Given : the time-stepper Xk+1 = Φτ(Xk), its linearized counterpart, and an initial guess +X0 for the fixed point. +For j = 1, 2, · · · , m +1. Compute the residual of the nonlinear equation r = Φτ(Xj) − Xj +2. Check for convergence. If ∥r∥ ≤ ε, return Xj as the solution. +3. If not converged, compute the Newton correction by solving Jx = −r +4. Update the solution as Xj+1 = Xj + x. +The linear system in step 3 of this iteration is typically solved using a GMRES solver (or other +Krylov-based solvers such as BiCGSTAB [268] or IDR [242]) making use of the previously imple- +mented Arnoldi factorization (the matrix to be factorized being J). The GMRES procedure is +presented in Algorithm 3. It should be emphasized that the linear equation in step 3 does not +need to be solved with high precision at each step. It is indeed sufficient to ensure that the Newton +correction x reduces the norm of the residual r = Φτ(Xj+1)−Xj by one or two orders of magnitude +by setting the tolerance in the iterative linear solver to e.g. 0.01∥r∥. Although this may increase the +20 + +number of Newton steps before convergence is reached, each iteration is computationally cheaper +and faster as fewer Krylov vectors need to be generated by the iterative linear solver, thus reducing +the overall time to solution. Hereafter, such a strategy is referred to as Newton-Krylov with dynamic +tolerances. +21 + +Algorithm 3 – Solving Ax = b with GMRES +Consider the linear system Ax = b with A ∈ Rn×n, and both x and b ∈ Rn. We will +assume furthermore that n ≫ 1 so that solving this system with standard direct solvers (e.g. +LU factorization) is intractable. One of the most famous iterative methods to solve such +large-scale systems is the generalized minimal residual method (GMRES), introduced +by Yousef Saad and Martin H. Schultz [222] in 1986. Based on the Arnoldi factorization, it +iteratively approximates the solution to Ax = b by the vector xk in a k-dimensional Krylov +subspace with minimal residual. This task can be formulated as the following optimization +problem +minimize +xk∈K +∥b − Axk∥, +where K(r0, A) is a k-dimensional Krylov subspace iteratively constructed using the residual +r0 = b − Ax0, with x0 our initial guess for the solution (often taken as the zero vector), and +∥ · ∥ denoting the Euclidean norm. Consider now the k-step Arnoldi factorization +AVk = Vk+1‹ +Hk, +with Vk ∈ Rn×k an orthonormal matrix (i.e. VT +k Vk = Ik), and ‹ +Hk the resulting (k + 1) × k +upper Hessenberg matrix. The unknown vector xk can be expressed as +xk = x0 + Vky, +with y ∈ Rk an unknown low-dimensional vector. The columns of Vk+1 being orthonormal, +introducing this expression into the GMRES objective function yields +∥b − Axk∥ = ∥b − A (x0 + Vky) ∥ += ∥r0 − AVky∥ += ∥βv1 − Vk+1‹ +Hky∥ += ∥Vk+1 +Ä +βe1 − ‹ +Hky +ä +∥ += ∥βe1 − ‹ +Hky∥, +with e1 the first vector in the standard basis of Rk+1, and β = ∥r0∥. The low-dimensional +vector y is thus a solution to the following least-squares problem +minimize +y +∥βe1 − ‹ +Hky∥, +whose solution is given by y = β‹ +H† +ke1, where ‹ +H† +k = +Ä‹ +HT +k ‹ +Hk +ä−1 ‹ +HT +k is the Moore-Penrose +pseudoinverse. If the residual is too large, a new Krylov vector is generated following the +Arnoldi procedure and the iteration continues until ∥βe1 − ‹ +Hky∥ is small enough. Numerous +variants of GMRES exist, most notably when A is ill-conditioned. For more details, inter- +ested readers are referred to the excellent book Iterative methods for sparse linear systems +by Y. Saad [223]. +Let us now consider a critical point in the above formulation of the problem, namely its con- +vergence and computational cost. For a reasonable initial guess, the number of Newton iterations +is expected to scale as O(k) where k is the number of eigenvalues of Mτ in the vicinity of the +22 + +unit circle (see [138] for a discussion about the convergence properties). +In typical production +runs, only a handful of eigenvalues may be unstable or close to being unstable. In our numerical +experiments, the Newton solver usually converges in more or less ten iterations, irrespective of the +discretization of the underlying partial differential equation. From a computational point of view, +the cost of each Newton iteration is, however, dominated by the call to the GMRES solver where +each new Krylov vector is obtained from a linearized simulation over the integration time τ. This +parameter τ plays a crucial role in the number of Krylov vectors that need to be generated to +achieve convergence as it directly impacts the eigenvalue distribution of the exponential propagator +Mτ = exp(τL) and thus of the Jacobian J = Mτ − I. As τ increases, the gap between the leading +eigenvalues and the others increases, and GMRES requires fewer iterations to converge (and thus +fewer Krylov vectors must be stored in memory). The wall-clock time of each GMRES iteration +however increases. Still, a major advantage of the time-stepper formulation is that it does not +require preconditioning strategies to perform well (although it can benefit from them, as shown in +[82, 263] and the discussion in section 5.1). A short parametric study was performed to evaluate the +performance and sensitivity of the method with respect to the size m of the Krylov subspace and +the integration time τ. At least three runs4 are performed with m ∈ (50, 75, 100, 125, 150, 175, 200) +and τ = (T/12, T/10, T/8, T/6, T/4, T/2, T), where T is the characteristic timescale of the leading eigen- +value. This is illustrated in figure 5 for two different flow cases: the two-dimensional cylinder flow +at Re = 80 and the two-dimensional open cavity flow at Re = 4700. For the computation of the +leading eigenvalues, we can see that similar times to solution are obtained for +4 < mτ +T +< 20, +i.e. the product of the dimension of the Krylov subspace and the sampling time τ is sufficient to +cover between 4 and 20 periods of the instability. For the cases of non-oscillatory instabilities, e.g. +from a pitchfork bifurcation, we suggest a sampling period of τ = 1 non-dimensional time unit as +a starting point if no estimate of the doubling time of the instability is available. For the cylinder +flow, mτ = 67 using dynamic tolerances resulted in a minimum total computation time of about +a minute, while mτ = 1128 led to a computation time of over 16 minutes. For the open cavity +2D flow, the minimum total computation time of 30 seconds was achieved with mτ = 10, and the +maximum time over 6 minutes was calculated with mτ = 120. +3.2.2 +Periodic orbit computation +Periodic orbits are solutions to +X = Φτ(X) +for +τ = τ ∗, +(41) +where τ ∗ is the period of the orbit. They are the roots of +F(X, τ) = Φτ(X) − X. +(42) +The above system of equations is underdetermined: it has only n equations for n + 1 unknowns +(the last unknown being the period of the orbit). To close the system, an extra phase condition +must be considered to select a particular point on the orbit (see Algorithm 4). Various possibilities +have been suggested in the literature. For example, in AUTO [84, 85] an integral constraint is used. +4The tests were computed in a single node of the Jean Zay HPE SGI 8600 supercomputer with two Intel Cascade +Lake 6248 processors (20 cores at 2.5 GHz) with the Intel Compiler version 2020.4 More information in http: +//www.idris.fr/eng/jean-zay. +23 + +Figure 5: Time to solution (in CPU minutes) versus different pairs of Krylov basis size m and +integration time τ for fixed point computation: (a) base flow of 2D flow past a circular cylinder at +Re = 80, where the initial condition is assumed to be the base flow at Re = 40; (b) base flow of the +2D open-cavity at Re = 4700 starting from the base flow at Re = 4000. Black markers represent +time to solution computed with fixed solver tolerances set to 10−10. Red markers represent cases +where the tolerance was tightened after each Newton step, from 10−5 at the first step to the target +value 10−10 at the final step. The parametric study is carried out using an automated Python script +(found in validations/newton_loop/autorun.py) that loops over previously defined ranges of m +and τ. +Here, a simpler condition is used. Given an initial condition X0 = X(0), the phase condition is +chosen as follows +F(X0) · (X − X0) = 0, +(43) +where F(X0) is the time-derivative of the system evaluated at X0. A solution to this bordered +system +® +Φτ(X) − X = 0, +F(X0) · (X − X0) = 0, +(44) +can be obtained using a Newton-Krylov solver. The Jacobian of this system is +J = +ïMτ − I +F (Φτ(X0)) +FT (X0) +0 +ò +. +(45) +As before, the corresponding matrix-vector product requires a single call to the linearized time- +stepper (which includes many smaller time steps) to evaluate Mτx (see the upper left block of the +Jacobian matrix). Evaluation of the other terms should be readily available. +24 + +(b) +(a) +101 + minutes] +[CPU +100 +t +101 +102 +102 +101 +mT/T +mT/TAlgorithm 4 – Newton-Krylov solver for periodic orbits +Given : the time-stepper Xk+1 = Φτ(Xk), its linearized counterpart, and an initial guess +X0 for the fixed point. +For j = 1, 2, · · · , m +1. Compute the residual of the nonlinear equation r = Φτ(Xj) − Xj +2. Check for convergence. If ∥r∥ ≤ ε, return Xj as the solution. +3. If not converged, compute the Newton correction by solving +ïMτ − I +F (Φτ(X0)) +FT (X0) +0 +ò += +ï x +∆τ +ò +. +4. Update the solution as Xj+1 = Xj + x, τj+1 = τj + ∆τ. +When evaluating the matrix-vector product Mτx, both the original nonlinear system and the +linearized one need to be marched forward in time. While this increased computational cost is +limited for small-scale systems, it may become quite significant for large-scale systems. A simple +strategy to alleviate this is to precompute the tentative orbit Xk(t) for t ∈ [0, τk] at the beginning +of each Newton iteration and store all time steps in memory. Then, only the linearized system needs +to be marched forward in time with its coefficients updated at each time step. If one is memory- +bounded, only a limited number of time steps of the nonlinear system can be stored in memory +and the intermediate steps can be reconstructed using for instance cubic spline interpolation or +temporal Fourier interpolation. +The Newton-Krylov algorithm presented herein corresponds to the standard shooting method. +It is by far the simplest method to implement for the computation of periodic orbits in large-scale +systems. Other techniques exist. For instance, multiple shooting [225] leverages the concept of +a Poincar´e section while temporal collocation transforms the orbit computation into a boundary +value problem. Recently, Shaabani-Ardali et al. [237] have also adapted ideas from feedback control +with time delays to stabilize unstable periodic solutions. In figure 6, we show the evolution of the +residual with respect to the computational time for stabilization of a periodic base flow with an +imposed frequency using the time-delayed technique and the proposed Newton GMRES. We can +observe a striking difference in the residual deflation when comparing the two techniques. +3.3 +Large-scale eigensolvers +Having computed a fixed point or a periodic orbit, one is often interested in its stability properties. +These can be its asymptotic stability (i.e. modal stability characterized by the eigenvalues of the +Jacobian matrix L) or short-time stability (i.e. non-modal stability characterized by the singular +values of the exponential propagator exp(τL)). +As discussed in section 2.2.1, given a fixed point X∗, the linearized dynamics are governed by +dx +dt = Lx, +(46) +where L is the evolution operator linearized about X∗. For a periodic orbit, these dynamics are +25 + +Figure 6: Residual deflation as a function of the total computation time (both computed in the +same hardware) for time-delayed feedback (open gray circles) and Newton GMRES (filled black +diamonds) for the harmonically forced jet presented in section 4.2. To obtain a continuous signal +over time for the Newton method, we sum the number of calls to the linearized solver between each +Newton iteration. We observe decays proportional to t−2 for the time-delayed feedback and t−14 +for the Newton. +governed by +dx +dt = L(t)x, +(47) +with L(t+τ) = L(t) and τ the period of the orbit. In a time-stepper formulation, these continuous- +time linear systems are replaced by the following discrete-time one +xk+1 = Mτxk, +(48) +with Mτ being the exponential propagator or monodromy matrix, depending on the context. The +matrix-vector product Mτxk amounts to integrating forward in time the linearized equations. While +for a fixed point, Mτ and L have the same set of eigenvectors, their eigenvalues are related by +λi = log(µi) +τ +, +(49) +where τ is the sampling period. For a periodic orbit, the eigenvalues µi of Mτ are directly the +Floquet multipliers needed to characterize the stability of the solution. +In both cases, the leading eigenpairs of Mτ can easily be computed using the Arnoldi factor- +ization described in section 3.1 and algorithm 5. Consider the factorization +MτV = VH + βeT +mr, +(50) +with V ∈ Rn×m an orthonormal Krylov basis, H ∈ Rm×m an upper Hessenberg matrix and r the +unit-norm residual after m steps of the Arnoldi iteration. Introducing the ith eigenpair (ˆµi, yi) of +the m × m upper Hessenberg matrix into the Arnoldi factorization leads to +∥MτVyi − Vyiˆµi∥2 = |βeT +myi|. +(51) +Hence, if the left-hand side is small enough, the pair (ˆµi, Vyi) provides a good approximation of the +ith eigenpair of the operator Mτ. If one is interested in the short-time stability properties instead, +26 + +10-2 +10-4 +10-6 +E +10-8 +10-10 +10-12 +10-2 +10-1 +100 +101 +102 +CPU hoursAlgorithm 5 – Solving Ax = λx with the Arnoldi factorization +Consider the eigenvalue problem +Ax = λx, +with A ∈ Rn×n, x ∈ Cn and λ ∈ C. The k-step Arnoldi factorization reads +AVk = VkHk + βvk+1eT +k , +with Vk ∈ Rn×k an orthonormal matrix (i.e. VT +k Vk = Ik), Hk the k × k upper Hessenberg +matrix, and ek the kth vector in the standard basis of Rk. Given the ith eigenpair (λi, yi) of +the Hessenberg matrix, the ith eigenpair of the original matrix A can be approximated as +λi, +and +xi = Vkyi, +with the low-dimensional eigenvector yi being normalized such that ∥yi∥2 = 1. The residual +associated to this approximate eigenpair (also known as a Ritz eigenpair) is given by +∥Axi − λixi∥ = ∥AVkyi − λiVkyi∥ += ∥Vk (Hk − λiI) yi + vk+1βeT +k yi∥ += |β||eT +k yi|. +If this residual is small enough for a sufficiently large number of Ritz eigenpairs, then the +computation stops. Otherwise, a new Krylov vector is added to the basis V, and the Arnoldi +factorization continues until the desired number of Ritz eigenpairs have converged below the +user-defined tolerance. +the leading singular modes and associated gain can be computed using the same algorithm where +Mτ is replaced by M† +τMτ = exp�τL†� exp (τL). +As for the Newton-Krylov solver presented in section 3.2, the computational time is dominated +by that of the call to the linearized solver needed to compute the matrix-vector product Mτxk. For +fixed points, the choice of τ is also important as it plays a key role in the spectral gap between the +eigenvalues of interest and the others. This is discussed in section 5.2. The initial vector v1 used to +generate the Krylov subspace is also important. Assuming random white noise distributed in the +perturbation fields, the eigenpairs effectively start to converge only after a sufficiently large number +of Krylov vectors have been generated such the transients are washed out of the computational +domain. The number of Krylov vectors that must be generated before this happens is, of course, +dependent on the sampling period τ and the size of the computational domain under consideration. +From our experiences, assuming only one eigenvalue is unstable a good trade-off in terms of time- +to-solution and computational cost is obtained when one chooses the size m of the Krylov subspace +and the sampling period τ such that +mτ +T += O(10), +where T is an a priori estimate of the typical timescale of the instability. If multiple eigenvalues +are unstable, T can be selected as the slowest time-scale. We point out that such estimates should +be regarded as indicative only, since effective computational performance depends on many other +parameters not considered here (hardware, operating system, compiler, etc.). Figure 7 shows a +parametric study performed to evaluate the performance and sensitivity of the method with respect +27 + +to the size m of the Krylov subspace and the integration time τ. At least three runs were made +with m ∈ (50, 75, 100, 125, 150, 175, 200) and τ = (T/12, T/10, T/8, T/6, T/4, T/2, T), where T is the +characteristic timescale of the instability. Two cases are considered: the two-dimensional cylinder +flow at Re = 80 and the two-dimensional open cavity flow at Re = 4700. +Figure 7: Time to solution (in CPU minutes) versus different pairs of Krylov base size m and +integration time τ for the computation of eigenvalues: (a) 2D flow past a circular cylinder at +Re = 80 (T = 1/0.125); (b) 2D open-cavity at Re = 4700 (T = 1/1.676). Red markers represent +the time to solution for the convergence of 4 eigenmodes and black markers are for 40 eigenmodes. +For each value of mτ three runs with different initial conditions are computed to account for +fluctuations of the computer. Eigenvalues with residual lower than 10−6 are considered converged. +The parametric study is carried out using an automatic python script (located in validations/ +eigen_loop/autorun.py) looping over previously defined ranges of m and τ. +4 +Examples +This section illustrates different applications of the use of Krylov methods to study large-scale +dynamical systems. All examples are taken from fluid dynamics applications. Numerical simula- +tions rely on the spectral element solver Nek5000 [80, 100] and the dedicated open-source toolbox +nekStab nekstab.github.io (see Appendix A). It should be noted that, although we have focused +our attention on a particular CFD solver, the methods presented earlier are quite general and can be +relatively easily implemented in other partial differential equation solvers. We give a brief physical +description of each case, as well as details of the base flow and stability calculation, and a compar- +ison with a reference work from the literature. Finally, a brief bifurcation analysis is presented for +each case. All the files needed to run these examples can be found in the nekStab/examples folder, +available in the repository github.com/nekStab. +4.1 +The flow in a two-dimensional annular thermosyphon +Under the influence of unstable thermal stratification, and for a range of control parameters, the +two-dimensional flow in an annular thermosyphon is perhaps one of the simplest and cheapest +computational test cases. +The geometry considered is the same as in [157]. +It consists of two +concentric circular enclosures, the inner radius being R1 and the outer radius R2. The ratio of the +28 + +(b) +(a) +102 +101 +[CPU +t +100 +100 +102 +102 +101 +101 +mT/T +mT/Touter to inner radius is set to +R2 +R1 += 2. +A constant temperature T0 is set at the upper walls, while the lower ones are set at a temperature +T1 = T0 + ∆T, with ∆T > 0. Hereafter, we work with the non-dimensional temperature ϑ(x, y) +defined as +ϑ(x, y) = T(x, y) − T0 +∆T +, +where x and y are the horizontal and vertical coordinates. The origin of our reference frame is +chosen to be the center of the thermosyphon. Using this non-dimensionalization, the temperature +at the lower walls is thus ϑw(y < 0) = 1, while the temperature at the upper ones is ϑw(y > +0) = 0. Gravity acts in the vertical direction, along −ey, and is characterized by the gravitational +acceleration g. +Assuming the working fluid is Newtonian, it is characterized by its density ρ, its dynamic +viscosity µ, its thermal expansion coefficient β, and its thermal diffusivity α. Using ∆T as the +temperature scale, and R2 −R1 as the length scale, we can define two non-dimensional parameters, +namely the Rayleigh number +Ra = ρgβ∆T (R2 − R1)3 +µα +, +and the Prandtl number +Pr = ν +α, +where ν = µ/ρ is the kinematic viscosity. For this example, the Prandtl number is set to Pr = +5, and the Rayleigh number is varied. For simplicity, the flow is assumed two-dimensional and +incompressible, so that the effect of density variations due to temperature can be modeled using +the Boussinesq approximation. Under these assumptions, the dynamics of the flow are governed +by the following Navier-Stokes equations +∂u +∂t + ∇ · (u ⊗ u) = −∇p + Pr∇2u + RaPrϑey, +∂ϑ +∂t + (u · ∇) ϑ = ∇2ϑ, +∇ · u = 0, +where u(x, t) is the velocity field, p(x, t) the pressure field, and ϑ(x, t) the temperature field. +The computational domain is discretized using 32 spectral elements uniformly distributed in the +azimuthal direction, and 8 elements uniformly distributed in the radial direction. Within each +element, Lagrange interpolation of order N = 7 based on the Gauss-Lobatto-Legendre quadra- +ture points is used in each direction, resulting in 16 384 grid points. +Temporal integration is +performed using a third-order accurate scheme, and the time step has been chosen to satisfy the +Courant–Friedrichs–Lewy (CFL) condition with Courant number Co < 0.5 for all the simulations. +Despite the lack of turbulent dynamics due to the absence of the vortex stretching mechanism, the +flow configuration can exhibit Lorenz-like chaotic dynamics. This low-cost case follows the same +structure as the bifurcation diagram of the Lorenz system. +4.1.1 +Pitchfork bifurcation +The first bifurcation of the flow occurs at the critical Rayleigh number Rac,1 ≃ 494. It is associated +with symmetry breaking in the temperature distribution, which leads to the emergence of a station- +ary convection cell. The calculations presented were performed with τ = 1 (diffusive time unit) and +29 + +Figure 8: Eigenvalues for the destabilization of the fixed point for the thermal convection on an +annular thermosyphon. +Figure 9: The flow in a two-dimensional annular thermosyphon: (a) temperature field (ϑ) of the +bilateral symmetric fixed point and (b) real part of the temperature field (Re(ϑ)) of the unstable +steady mode at Ra ≃ 499. +a Krylov subspace of dimension m = 120. The choice of the sampling period for stationary modes is +not straightforward, as a small value of τ leads to a poorly conditioned basis and the generation of +spurious modes, while a large sampling period leads to an excessive computational cost. The base +flow, corresponding to pure conduction, is shown in figure 9(a). The eigenvalue spectrum in fig- +ure 8 shows a purely real eigenvalue stepping into the upper half-complex plane for Ra ≥ 494. The +associated eigenvector is depicted in figure 9(b) and leads to symmetry breaking. The correspond- +ing bifurcation is thus a pitchfork bifurcation. Above Rac,1, the conduction-dominated symmetric +base flow is no longer stable and is replaced by a convection cell as shown in figure 11(a). From +symmetry considerations, this convection cell is equally likely to be associated with clockwise or +counterclockwise flow (as shown in figure 11(a)). +4.1.2 +Hopf bifurcation +The new base flow is stable over a wide range of Rayleigh numbers. It eventually becomes un- +stable at Rac,2 ≃ 16 081 through a Hopf bifurcation. +This is indicated by a pair of complex +conjugate eigenvalues of the associated linearized Navier-Stokes operator moving toward the upper +half-complex plane in figure 10. The spatial structure of the leading mode is shown in figure 11(b). +30 + +0.2 - +Ra = 510 +0.0 +6 +Ra = 499 +A +Ra = 490 +-0.2 - +Ra = 480 +口 +-0.3 +-0.2 +-0.1 +0.0 +0.10.0 +0.5 +1.0 +0 +(a) +(b) +2 +90 +-2 +-2 +0 +2 +-2 +0 +2Figure 10: Eigenvalues of the flow of the steady convection cell. The dashed line represents the +frequency f = 7.67 from a DNS at Ra = 16100. +Figure 11: The flow in a two-dimensional annular thermosyphon: (a) temperature field (ϑ) of an +unstable steady convection cell and (b) real part of the temperature field (Re(ϑ)) of the unsteady +unstable mode at Ra = 16100. +The convection cell shown in figure 11(a) starts to oscillate with a characteristic Strouhal number +St = 7. The frequency predicted by the linear analysis and the frequency measured in our DNS +match very well near the bifurcation point when the nonlinear distortion of the base flow is minimal. +Eigenvalue calculations were performed with τ ≈ 0.014 (corresponding to a standard recom- +mendation for the sampling period of τ = T/8) convective time and using a Krylov subspace of +dimension m = 120. Base flows were calculated using the Newton-Krylov method under the same +conditions. For comparison metrics, the calculation of the unstable base flow at Ra = 16 100 start- +ing from the base flow at Ra = 16 000 took 147 seconds with Newton-Krylov compared to 1071 +seconds with selective frequency damping (SFD) [1] (i.e. one seventh of the time). For SFD, we +considered a parameterization proposed by [61] that leads to a more robust selection of the cutoff +and gain for low-pass filtering compared to the original guidelines [1]. +31 + +0.10 +- +0.05 +Ra = 16100 +0.00 +b +Ra = 16082 +A +-0.05 +Ra = 16050 +Ra = 16000 +口 +-0.10 +口 +-10 +0 +5 +10 +-5 +f0.0 +0.5 +1.0 +0 +(a) +(b) +2 +9o +-2 +-2 +0 +2 +-2 +0 +24.2 +The harmonically forced jet +Our attention now shifts towards a time-periodic flow harmonically forced via the inflow boundary +condition. The dynamics of the flow are governed by the incompressible Navier-Stokes equations +∂u +∂t + ∇ · (u ⊗ u) = −∇p + 1 +Re∇2u, +∇ · u = 0. +(52) +The Reynolds number is defined as +Re = U0D +ν +, +where U0 is the velocity at the jet centerline, D the jet diameter, and ν the kinematic viscosity of the +fluid. For simplicity, the jet is assumed to be radially symmetric, and the axisymmetric formulation +of the Navier-Stokes is considered with x = (z, r), z representing the streamwise direction and r +the radial direction. The time-periodic structure is forced via a Dirichlet inflow boundary condition +prescribed as +u(z = 0, r, t) = 1 +2 +ß +1 − tanh +ï +1 +4θ0 (r − 4r−1) +ò™ +(1 + A cos(ωft)), +with A = 0.05 the force amplitude, θ0 = 0.025 the initial thickness of the dimensionless shear layer, +and angular frequency ωf. The non-dimensional frequency is the Strouhal number +St = ωfD/(2πU0). +The computational domain extends from 0 to Lz = 40 in the streamwise direction and 0 to Lr = 5 +in the radial direction. +The domain is discretized with nz × nr = 160 × 30 spectral elements. +Lagrange interpolants of order N = 5 are considered, resulting in 172 800 grid points. Based on +the recent work of [238], we reproduced a case with St = 0.6 and investigated its modal stability +properties. Under these conditions, a forced limit cycle is formed. For subcritical conditions (i.e. +Re < Rec), vortices form periodically along the shear layer. They are then transported in the +streamwise direction before fading out because of viscous diffusion. No pairing phenomena are +identified despite the appearance of harmonics in the velocity signals. Above the critical Reynolds +number Rec = 1371, the vortices spontaneously start to pair, forming larger vortices. The vortex +pairing is connected to a subharmonic instability created via a period-doubling bifurcation. +4.2.1 +Period-doubling bifurcation +In [238], the time-delayed feedback technique [237] was used to stabilize periodic orbits. +The +technique is based on the nonharmonic component filtering approach introduced by [210] using an +optimal filter gain derived in [237]. Here, the (unstable) time-periodic base flow is computed using +the Newton-Krylov method with dynamic tolerances (introduced in section 3.2.1), with τ = 1/St +and a Krylov subspace dimension of m = 128 until a residual level of 10−11. The same parameters +are used for the Floquet analysis, which is sufficient to converge 20 eigenpairs to a precision of +10−6. An unstable base flow without vortex pairing can be seen in figure 12(a). Figure 12(b) +illustrates the dominant Floquet mode. Figure 13 shows the spectrum of Floquet multipliers, with +the leading one leaving the unit circle along µ = −1, which is characteristic of a period-doubling +bifurcation. The critical number Rec,1 ≃ 1371.18 is in excellent agreement with the reference value +Rec,1 ≃ 1371 given in [238]. Despite the strong non-normality of the system operator, the Floquet +32 + +Figure 12: The harmonically forced jet: (a) vorticity component of the stabilized unstable limit +cycle at supercritical Re = 2000 and (b) spatial distribution of the subharmonic Floquet mode. +Inflow forcing with 5% amplitude and StD = 0.6. +analysis accurately predicts the leading mode responsible for the vortex pairing mechanism observed +in nonlinear simulations [238]. +We verified the predictions with nonlinear simulations at subcritical Re = 1370 and supercritical +Re = 2000, as shown in figure 14. At Re = 1370 (in black), the time series of the velocity probe and +its Fourier spectrum are characterized by the forced frequency St = 0.6 and its harmonics formed +by nonlinearities. At Re = 2000 (in red), one can see the response of the flow in the velocity +signal with the sharp increase in the subharmonic frequency at St = 0.3, showing the growth of +the secondary instability through a period-doubling bifurcation and the increase in the period of +the limit cycle. +4.3 +The flow past a circular cylinder +Let us now consider the example of a canonical cylinder flow assumed to be infinite in the spanwise +direction. The dynamics of the flow are governed by the Navier-Stokes equations (52), with the +Reynolds number defined on the basis of the free-stream velocity and the cylinder’s diameter. The +two-dimensional mesh considered in section 4.3.1 is made of 1464 spectral elements (66 in the flow +direction and 30 in the vertical), all of which are comparable to previously reported domains. To +limit the computational cost, we consider Lagrange interpolants of order N = 5, which shows +good convergence compared to N = 7 which leads to a total of 52 704 grid points. For the three- +dimensional problem considered in section 4.3.2, this mesh is extruded in the third direction, using +10 elements in the spanwise direction to a length Lz = 2π/βc ≃ 3.964. The total number of grid +points is then 3 162 240. As for the other examples, we consider a third-order accurate temporal +scheme and a time-step that satisfies Co < 0.5 for all simulations. +4.3.1 +Primary instability and sensitivity analysis +At low Reynolds numbers, the flow is two-dimensional, steady and symmetric with respect to the +cross-stream direction. According to Jackson [131] and more recently Kumar & Mittal [144], the +flow is expected to become unstable at Rec,1 = UD/ν ≈ 46.6, leading to the emergence of the +well-known von K`arm`an vortex street characterized by Stc = 0.125. This primary instability is a +canonical example of a supercritical Hopf-type bifurcation. Numerous wake flows [206] exhibit an +instability that leads to the onset of periodic vortex shedding and their characterization as flow +oscillators. +33 + +(a) +0 +2 +3 +5 +0 +(b) +2 +0 +-3 +3 +1 +0 +0 +5 +10 +15 +20 +25 +30 +35 +40 +ZFigure 13: Evolution of Floquet multipliers of the harmonic forced axisymmetric jet. The single +leading unstable eigenmode is associated with the vortex pairing mechanism and a period-doubling +bifurcation. The almost superposed black star represents the reference value from [238]. +Figure 14: Radial velocity signal at x, r = (5, 0.5) at subcritical Re = 1370 (black) and supercritical +Re = 2000 (red): (a) signal evolution and (b) discrete Fourier transform (DFT) spectrum; (c) +Phase portrait of the system in coordinates (v, ˙v, ¨v): one can observe the departure of the system +trajectory from the stable limit cycle (in black), timidly exploring the phase space before settling +into a period-doubled orbit (in red). +34 + +Re = 1300 +Re = 1370 +Re = 1375 +Re = 2000 +-1 +0 +1(a) +(c) +0.4 +0.2 +2 +0.0 +-0.2 +20 +40 +60 +80 +0 +100 +120 +140 +七 +(b) +105 +St1 =0.60 +-0. +St2 =0.30 +103 +-0.1 +2 +0.0 +101 +0.1 +10 +0.2 +10-3 +0.3 +2 +1 +3 +5 +0 +1.5 +2: +-1.0 +S +0.4 +-0.5 +0.0Sensitivity to base flow changes based on the linear Navier-Stokes operator was introduced by +Bottaro et al. [37] in a local framework and was later extended to the global framework by Marquet +et al. [174]. Base flow sensitivity analysis has been shown to provide valuable information for shape +optimization or actuator placement. The sensitivity analysis of the complex eigenvalue λ = σ + iω, +with the real part σ being the growth rate of the eigenmode and the imaginary part ω its frequency, +allows the computation of vector fields that highlight the net effects of generic small-amplitude base +flow modifications δU. Both the sensitivity analysis for generic modifications of a base flow (e.g. +U) and for a steady force (e.g. F) have been implemented in nekStab. +Structural changes in the complex eigenvalue δλ due to small-amplitude arbitrary base flow +modifications δU can be formally related through the inner product +δλ = ⟨∇Uλ|δU⟩. +The gradient ∇Uλ is a complex vector field that defines the sensitivity to base flow modifications, +given by +∇Uλ = −(∇u)H · u† + ∇u† · ˆu∗, +(53) +with the superscript H representing the conjugate transpose, † the adjoint, and ∗ the complex +conjugate (for a complete derivation, the reader is referred to Marquet et al. [174]). +The first +term of equation (53) is the sensitivity to transport modifications, while the second term is the +sensitivity to production modifications. Figure 15 depicts the real and imaginary parts of ∇Uλ. +These represent the sensitivity of the growth rate of the eigenvalue to base flow modifications, +and the frequency sensitivity of the eigenvalue to base flow modifications, respectively. Generic +base flow modifications in negative sensitivity zones promote stabilization of the eigenvalue (i.e. +reduction of the growth rate or frequency), whereas changes in positive sensitivity zones promote +destabilization or frequency increase. +Analogously, the sensitivity to a steady force can be derived by introducing an inner product +in relation to a volume force F in the form +δλ = ⟨∇Fλ|δF⟩. +The complex sensitivity function ∇Fλ is given by the knowledge of adjoint base flow fields in the +form +∇Fλ = U†. +(54) +This adjoint base flow is a solution to the linear system of equations +−∇U† · Ub + (∇Ub)T · U† − ∇P † − 1 +Re∇2U† = ∇Uλ, +∇ · U† = 0. +(55) +Assuming the adjoint Jacobian matrix L† has already been projected onto a divergence-free sub- +space, this equation can be written as +L†U† = ∇Uλ. +Note that, in a time-stepper framework, we do not have access to L†, but only exp�τL†�. Accounting +for the time derivative, the general solution of the linear system above is given by +U†(τ) = exp +Ä +τL†ä +U†(0) + +� τ +0 +exp +Ä +(τ − t)L†ä +∇Uλ dt. +35 + +Figure 15: Streamwise component of the sensitivity to base flow modifications ∇Uλ of the leading +eigenvalue λ at Reynolds number Re = 50. Spatial distribution of (a) the growth rate sensitivity +∇Uσ and (b) the frequency sensitivity ∇Uω. +In a time-stepper framework, equation (55) is thus replaced with +Ä +I − exp +Ä +τL†ää +U† = +� τ +0 +exp +Ä +(τ − t)L†ä +∇Uλ dt, +(56) +which is solved using the GMRES solver discussed earlier in the manuscript. Note that the right- +hand side is computed by running an adjoint simulation with the steady force ∇Uλ for τ time units +with a zero initial condition. Figure 16 depicts the real and imaginary parts of ∇Fλ. These are the +sensitivity of the growth rate of the eigenvalue to changes due to a steady force, and the associated +frequency sensitivity of the eigenvalue, respectively. These maps are consistent with those obtained +in [174] and experimentally in [246]. Although not reproduced here, the theoretical framework +was extended by Giannetti, Camarri, and Citro [109] to include sensitivity analysis with respect to +generic modifications and a force acting on periodic orbits. +4.3.2 +Pitchfork bifurcation +We now focus on the Floquet analysis of three-dimensional modes evolving in a two-dimensional +time-periodic base flow formed in the wake of a circular cylinder. This problem has been studied +in detail by Barkley & Henderson [19] on the basis of a Fourier expansion in the spanwise direc- +tion, as is the gold standard for demonstrating a (secondary) pitchfork bifurcation. In addition, +this symmetry-breaking bifurcation [189] is characterized by a single real eigenvalue that becomes +positive, corresponding to a single Floquet multiplier that leaves the unit cycle by µ = 1. +At +this point, the flow experiences a steady bifurcation (synchronous with the underlying periodic +36 + +Vuo +-0.40 0.18 +1. +9 +-1 +Vuf +(b) +2 +-0.40 0.18 +0- +9 +-1 +-2 +2 +0 +4 +6Figure 16: Normalized modulus of the sensitivity to a steady force ∇Fλ of the leading eigenvalue +λ at Reynolds number Re = 50. Spatial distribution of (a) the growth rate sensitivity ∇Fσ and +(b) the frequency sensitivity ∇Fω. +37 + +VFO +2 +0.1 +-1.1 +1 +9 +0 +-1 +VFf +(b) +2 +0.1 +-1.1 +0- +9 +-1 +-2 +2 +0 +4 +6orbit), thus not altering the temporal structure of the flow. The frequency of the underlying limit +cycle remains the same, but the spanwise invariance of the flow is broken, resulting in a three- +dimensionalization of the flow. Barkley & Henderson [19] report a secondary instability to what is +called mode A occurring at Rec,2 ≃ 188.5±1 with a critical wavenumber βc = 1.585 in the spanwise +direction, corresponding to a length of nearly four diameters in the spanwise direction. Using a +domain (Li, Lh, Lo) = −15D, ±22D, 25D, they report a limit cycle oscillating with St = 0.1954 at +Re = 190, and a leading synchronous Floquet mode with µ = 1.034. Later Giannetti, Camarri, and +Luchini [110], employing a finite element code and a domain (Li, Lh, Lo) = −16.5D, ±11D, 34.5D, +calculated Rec,2 ≃ 189.77 and some reference values at Re = 190 for the limit cycle frequency +St = 0.1971 and µ = 1.002 for mode A. Recently, Giannetti, Camarri, and Citro [109], using a +spectral element code and a domain (Li, Lh, Lo) = −15D, ±15D, 35D, obtained Rec,2 ≃ 189.71 and +reported a limit cycle with St = 0.1962 at Re = 190, as well as µ = 1.009 for mode A. +Barkley [21] mentions that accurate estimates of Floquet stability analyses can be obtained in +smaller domains (Li, Lh, Lo) = −8D, ±8D, 25D. Since nekStab relies on the spectral element solver +Nek5000, which does not use a Fourier decomposition in the spanwise direction, a finite-span domain +is specified. A limit cycle can be computed directly in the 3D domain, or due to the geometry of the +flow computational time can be saved simply by setting the spanwise velocity component to zero or +by extruding a 2D solution into a 3D mesh (e.g. using the pymech [54] package). In our case, using +the Newton-Krylov method for stabilizing UPOs, we first perform a DNS at subcritical Re = 187 +and thus transients from the initial condition are convected out of the computational domain, +obtaining a 2D subcritical periodic orbit in the 3D domain. The orbit period is measured from the +DNS and used as an initial guess for the Newton GMRES algorithm, together with a snapshot of +the orbit. In this way, we can gradually stabilize the UPOs at supercritical Reynolds numbers. A +Krylov subspace of dimension m = 64 is used, both for base flow and Floquet stability calculations. +Due to the almost degenerate nature of the eigenvalues, a Krylov-Schur iteration is used for the +convergence of at least 4 direct modes. The evolution of Floquet multipliers moduli is shown in +figure 17, as well as the best linear fit for their evolution along the Reynolds number. The fully 3D +leading unstable Floquet mode at Re = 190 is shown in figure 18 and shows excellent qualitative +agreement with the unstable mode presented in Barkley & Henderson [19] and also in Blackburn +and Lopez [34]. Our estimate for the critical Reynolds number Rec,2 = 189 using a domain with +βc = 1.585 agrees very well with the range of values reported by Barkley & Henderson [19] as well +as Giannetti, Camarri and Luchini [110], despite our necessity to fix a spanwise wavenumber that +could not precisely match the critical one reported, as well as a different mesh strategy, domain +size, and polynomial order. Specifically, our µ = 1.012 at Re = 190 differs by 2.15% with µ = 1.034 +by Barkley [21] and 1% with µ = 1.002 by Giannetti, Camarri, and Luchini [110]. +4.4 +The flow past side-by-side circular cylinders +The flow past two side-by-side cylinders is mainly governed by a Reynolds number based on the +cylinder diameter and the free-stream velocity (similar to a single-cylinder wake), but with the +addition of a separating distance (gap) between the surfaces. A mesh with 5092 elements and +polynomial order N = 7 is considered, spanning from -50 to 75 in the streamwise direction and +-50 to 50 in the vertical direction. Taking into account a gap g = 0.7, [59] reports a primary Hopf +bifurcation at Rec,1 ≃ 55 with a synchronized vortex shedding wake oscillating at Stc,1 = 0.11. At +Rec,2 ≃ 61.7, a secondary mode known as the “flip-flop” mode with Stc,2 = 0.02 becomes unstable +due to a supercritical Neimark-Sacker bifurcation. The flow is bistable: an asymmetric dual wake +arises, in which much slower switching can be observed. +38 + +Figure 17: Modulus of the dominant Floquet multiplier as a function of the Reynolds number for +the flow past a circular cylinder. +4.4.1 +Neimark-Sacker bifurcation +The Neimark-Sacker or secondary Hopf is characterized by the emergence of a new incommensurate +frequency in the flow. This bifurcation differs from a pitchfork when the instability created oscil- +lates with the same frequency of the periodic orbit and from period-doubling when the instability +created is subharmonic with respect to the periodic orbit. Figure 19 shows the spectrum of Floquet +multipliers for different Reynolds numbers. The critical Reynolds number is Rec,2 ≃ 61.17, which +is in excellent agreement with the reference value Rec,2 ≃ 61.6 reported in [59]. The mode leaves +the unit disk at an angle of 71 degrees (i.e. φ = tan−1(Im(µ)/ Re(µ))). Figure 20 shows a snapshot +of the vorticity distribution of the UPO under supercritical conditions Re = 67 and the vorticity +distribution of the unstable Floquet mode associated with the flip-flop mechanism. Figure 21(a,b) +show the time trace and the Fourier spectrum of the velocity recorded by a probe in the wake. The +main discrete Fourier transform (DFT) spectrum peak is located at the fundamental frequency +of the UPO, with another (second) peak occurring at the new incommensurable frequency. Fig- +ure 21(c) shows the phase-space representation of dynamics with the formation of a torus object +characteristic of quasiperiodic dynamics. +4.5 +Backward-facing step +The flow past a backward-facing step is chosen as an example of a transient growth analysis. We +reproduce the analysis presented in [35] at Re = 500 and take the step height as the charac- +teristic length scale. As before, the dynamics are governed by the incompressible Navier-Stokes +equations (52). We considered a mesh made up of 1670 elements, each locally discretized with a +polynomial order N = 5 in both directions. The domain spans from -10 to 50 in the streamwise +direction, and from -1 to 1 in the vertical direction. At Re = 500, the stationary base flow is +linearly stable. Yet, small perturbations can experience large transient growth due to the strong +non-normality of the linearized Navier-Stokes operator. The maximum gain envelope is calculated +and compared with the reference in figure 22, showing excellent agreement. The peak is located at +τ = 58, which corresponds to the maximum possible transient growth associated with the optimal +initial condition. Figure 23 shows the fixed point computed by the Newton-Krylov method, the +spatial distribution of the leading eigenvector of the direct-adjoint eigenproblem corresponding to +the optimal perturbation with maximum energy gain, and the optimal response at target time +39 + +1.04 +lμl(Re) ～ 0.008965Re - 0.694566 +Barkley & Henderson (1996) +Giannetti, Camarri and Citro (2019) +1.02 +Giannetti, Camarri and Luchini (2010) +1.00 +0.98 +187 +188 +189 +190 +ReFigure 18: The flow past a circular cylinder: semitransparent vorticity magnitude contours (ω = +0.35) of the limit cycle at supercritical Re = 190 superimposed with streamwise vorticity contours +(ωx = ±0.18) of the real part of the unstable steady mode. +τ = 58. Excellent agreement with reference [35] is obtained. +5 +Discussion +In this section, we discuss various practical and theoretical aspects of using a time-stepper formula- +tion. This includes the interpretation of time-stepping as an effective preconditioner in section 5.1 +and some observations on the convergence of leading eigenvalues for the linearized Navier-Stokes +operator in open shear flows in section 5.2. +5.1 +Time-stepper and preconditioning +Solving a linear system forms the most computationally intensive part of Newton solvers. Using a +standard approach, one needs to solve linear systems of the form +Lx = b, +where L ∈ Rn×n is the Jacobian matrix of the Navier-Stokes equations. +For the sake of this +discussion, we will assume furthermore that n is sufficiently large so that direct solvers cannot be +employed. No matter the discretization technique employed, this matrix tends to be ill-conditioned. +Consequently, directly solving the above system of equations with an iterative solver may lead to rel- +atively poor computational performances as the convergence rate of such solvers is directly impacted +by the conditioning of the matrix [223]. It should be emphasized furthermore that explicitly com- +puting this matrix-vector product might not moreover be easily accessible given a general-purpose +CFD solver. In order to overcome the first issue, practitioners typically use preconditioning, either +left preconditioning leading to a new system of the form +PLx = Pb, +or right preconditioning, leading to +LPy = b, +and +Py = x. +40 + +Figure 19: Evolution of Floquet multipliers for the flow past two side-by-side cylinders. A pair +of modes associated with the flip-flop instability leaves the unit cycle increasing its moduli (both +growth rate and frequency) as a function of Re in a Neimark-Sacker bifurcation. The black star +represents the reference value from [59] at Re = 67. +Figure 20: The flow through side-by-side circular cylinders: (a) snapshot of the limit cycle and (b) +Floquet “flip-flop” mode at Re = 67. +41 + +1.5 +X +LO +区 +X +C +Re = 60 +Re = 62 +Re = 63 +Re = 67 +-1.5 +-1.5 +0 +1.5 +R(μ)a +1.5 +2.5 +0.0 +0 +9. +-1.5 +-2.5 +(b) +0.5 +2.5 +0 +0.0 +9. +-0.5 +2.5 +0 +10 +15 +20 +5Figure 21: Vertical velocity signal at (x, y) = (5, 0) in the flow past side-by-side cylinders with +g = 0.7 for the subcritical Re = 60 (black) and supercritical Re = 62 (dark red) regimes: (a) signal +evolution and (b) DFT spectrum; (c) trajectory of the system in phase space using the coordinates +(i.e. v, ˙v) of both the limit cycle (black) and the torus (dark red). +Figure 22: Envelope of the optimal gain for the 2D backward-facing step problem computed for +Re = 500. +The parametric study is carried out using an automatic Python script (located in +examples/backward_facing_step/autocomp_tg.py) looping over a predefined range of τ. +42 + +(a) +(c) +0.1 +0.10 +2 0.0 +0.05 +-0.1 +0 +100 +200 +300 +400 +500 +600 +700 +800 +t +2 0.00 +(b) +105 +St1 =0.1113 +St1 =0.1125 +-0.05 +- +St2 =0.0206 +103 +0.10 +101 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.00 +0.02 +0.04 +0.06 +-0.06 +-0.04 +-0.02 ++S×104 +Barkley et al. (2008) +6 +2 +20 +40 +60 +80 +100 +0 +tFigure 23: The backward facing step problem: (a) fixed point at Re = 500, (b) optimal disturbance, +and (c) optimal response at the maximum amplification time horizon τ = 58. +In both cases, the matrix P ∈ Rn×n (known as the preconditioner) should be a reasonably good, +and more importantly cheap, approximation of the inverse of L. A judicious choice of P can lead +to a substantial reduction of the number of steps needed for the iterative solver to converge. This +choice however strongly depends not only on the spectral content of L but also on its structure +directly inherited from the discretization scheme employed. The most standard preconditioners +include Jacobi preconditioning, block LU, additive Schwarz methods, or algebraic and geometric +multigrids. In fluid dynamics, preconditioners designed specifically for fixed point computation +using a Newton solver include the Stokes and Laplace preconditioning by Tuckerman and col- +leagues [20, 45, 264, 265], or their extensions by Gelfgat [105]. Given a classical time-stepping +solver for the incompressible Navier-Stokes equations, these are relatively easy to implement (see +more discussions in [44]). Yet, to the best of our knowledge, no systematic methods are however +available to choose a priori the best preconditioner. +In contrast, the linear system involved in the time-stepper formulation of the Newton-Krylov +method reads +(exp(τL) − I) x = b, +where the sampling period τ plays a crucial role. At first, not much seems to be gained from this +formulation as, for a sampling period τ comparable to the discretization time step ∆t, we have +exp(τL) − I ≃ (I + τL) − I +∝ L. +Hence, for a small sampling period, the condition number of exp(τL) − I is directly proportional +to that of the Jacobian matrix L. However, in practice, τ is of the order of a few hundred or a few +thousand time-steps ∆t (i.e. τ ≫ ∆t) such that the first-order Taylor approximation of the matrix +exponential does not apply. +An upper bound on the number of iterations needed to drive the residual ∥rk∥2/∥r0∥2 below a +given tolerance ε can be derived analytically. Assume for the moment that all the eigenvalues of +L lie in the stable complex half-plane. Furthermore, we will assume that the leading eigenvalue λ1 +satisfies +Re(λ1) ≤ −δ, +with δ > 0. The spectrum of the matrix J = exp(τL) − I then satisfies +spec(J) ∈ D = {z : |z + 1| ≤ exp(−τδ)} . +43 + +(a) +9 +0 +(b) +0 +0 +(c) +90 +5 +15 +-5 +0 +10 +20 +25 +30 +35Adapting the derivation in [223] of the general purpose GMRES method to our particular case, it +can be shown that the number k of iterations needed to drive the normalized residual below a given +tolerance ε satisfies +ε ≤ κ(V) exp(−kτδ), +where V denotes the matrix of eigenvectors of J (which are identical to the eigenvectors of L), and +κ(V) = ∥V∥2∥V−1∥2 its condition number. From this expression, we can then write +k ≤ 1 +τδ log +Åκ(V) +ε +ã +. +It is well known that such upper bounds tend to be overly pessimistic, and typically are worst-case +scenarios. It nonetheless highlights the fact that increasing the sampling period τ is likely to reduce +the number of GMRES iterations needed for convergence. This is consistent with the fact that, if +L has only stable eigenvalues, then +lim +τ→∞ exp(τL) − I = −I. +Moreover, when exp(τL) has p eigenvalues outside of the unit disk, it can be shown that GMRES +only needs O(p) extra iterations to converge, and this upper bound remains unchanged. +Note +however that, as we increase τ, the computational cost associated to generating each Krylov vector +also increases, and a trade-off needs to be found between the computational cost of generating +a Krylov vector and the memory footprint of storing a large Krylov basis. It should be noted +furthermore that, while exp(τL) − I is well conditioned, this is obtained at the cost of taking many +small timesteps. In contrast, the operator in [20, 264, 265] is not as well conditioned, but consists +of a single timestep. This trade-off between fewer GMRES iterations for a well-conditioned but +costly operator vs. more GMRES iterations for a less well-conditioned but inexpensive operator +is discussed in detail in [263]. From a practical point of view, using a time-stepper formulation +can nonetheless be understood as an effective and easy-to-use preconditioning strategy as already +pointed out by Tuckerman and collaborators [20, 264, 265]. +5.2 +Observations about the convergence of eigenvalue computations +Let us now discuss some practical aspects associated with eigenvalue computations. These include +some general statements about the convergence of the leading eigenvalues, the choice of the sampling +period τ, as well as practical considerations regarding the influence of the streamwise extent of the +domain for open flows. +5.2.1 +General statements about the convergence of the Arnoldi iteration +Suppose the matrix exp(τL) ∈ Rn×n is diagonalizable, i.e. +exp(τL) = VDV−1, +with Dii = µi, and consider the orthonormal matrix Q ∈ Rn×k obtained after a k-step Arnoldi +factorization started with the unit-norm vector ˆb = +b +∥b∥2 . The vector ˜b can be expressed as the +linear combination of the eigenvectors such that +ˆb = +n +� +i=1 +αivi. +44 + +It can be shown that the approximation of the ith eigenvector vi in the Krylov basis Q satisfies +���� +Ä +I − QQT ä +vi +���� ≤ +Ü +n +� +j=1 +j̸=i +|αj| +|αi| +ê +ε(k) +i +. +(57) +A detailed derivation of this statement can be found in Appendix B. In the expression above, ε(k) +i +is given by +ε(k) +i += +min +p∈Pk−1 +p(µi)=1 +max(|p(µ1)|, · · · , |p(µn)|), +where Pk−1 denotes the set of polynomials of degree k − 1. Two key observations can be derived +from this upper bound. +Fast convergence for isolated eigenvalues +Suppose µi is an isolated eigenvalue such that the +rest of the spectrum can be enclosed in the disk +D = {z : |z − c| < ρ} , +i.e. a disk centered in c and of radius ρ. In a time-stepping framework, the center c typically is the +origin. Following [223], an upper bound for ε(k) +i +is given by +ε(k) +i +≤ κ(V) +Å +ρ +|µi − c| +ãk +, +where κ(V) ≥ 1 is the condition number of the matrix of eigenvectors. Clearly, the farther away µi +is from the rest of the spectrum, the smaller this upper bound, and hence the faster the convergence. +This observation is of utmost importance in numerous fluid dynamics applications where branches of +eigenvalues are pretty common. These branches are notoriously associated with the non-normality +of the linearized Navier-Stokes operator. Not only do they impact directly on the condition number +κ(V), but they are also non-isolated eigenvalues which are thus much harder to converge. +Influence of the starting vector +Note that ∥ˆb∥ = 1, and suppose the eigenvectors have been +normalized such that ∥vi∥ = 1. Then, the coefficients {αi}i=1,n satisfy +1 = ∥α1v1 + · · · + αnvn∥ ≤ |α1| + · · · + |αn|, +and let +ξi = +1 +|αi| +n +� +j=1 +|αj| − 1 ≥ +1 +|αi| − 1. +Suppose now that αj = δ for all j ̸= i, with δ small. Then, αi = O(1). Under these conditions, +the prefactor ξi is small, and the Arnoldi iteration exhibits a fast convergence for the ith eigenpair. +Alternatively, if αi is very small, then ξi ≫ 1, and we have slow convergence. This observation gave +rise to the rule-of-thumb that, for eigenvalue problems, the Arnoldi method will favor eigenvectors +having large components in the starting vector. Unless we have prior knowledge of the eigenvectors +we wish to compute, starting the Arnoldi iteration with a genuinely random vector increases the +probability that the eigenpairs of interest will converge. +45 + +Figure 24: Convergence history of the leading eigenvalue as a function of the total integration time +(defined as the product of the sampling period τ and the number m of Krylov iterations.) for the +flow past a circular cylinder at Re = 50 given different values of the sampling period τ: (a) the +domain extends from −16 to +50 in the streamwise direction and (b) the domain extends from −16 +to +100 in the streamwise direction. +5.2.2 +Influence of the domain extent for open shear flows +We now verify the influence of the streamwise domain length on the convergence of the leading +eigenpairs in open shear flows. We performed a series of computations for the flow past a circular +cylinder at Re = 50. In this case, the leading eigenpair corresponds to the von K´arm´an vortex street +mode associated with a supercritical Hopf bifurcation (as previously introduced in 4.3). Figure 24 +depicts the evolution of the leading eigenvalue residual as a function of the total integration time +T defined as the product of the number m of Krylov iterations and the sampling period τ, i.e. +T = mτ. Letting Lx be the streamwise extent of the computational domain and U∞ the velocity +scale, this figure highlights a rapid convergence of the leading eigenvalue once the total integration +time exceeds +mτ > Lx +U∞ +, +with U∞ = 1 in the non-dimensional case. This is easily explained by the fact that the usual choice +is to start from an initial perturbation vector consisting of noise (which aims to excite the entire +computational domain while minimizing any bias). The leading eigenvalue then starts to converge +after one flow-through time. +46 + +(a) +mT = 50 +10-1 +T二2 +- +T=1 +10-4 +T=0.5 +T = 0.25 +10-7 +E +- +10-10 +- +- +10-13 +- +I +10-16 +50 +100 +150 +200 +250 +300 +(b) +102 +mT = 100 +T=2 +10-1 +T=1 +T=0.5 +10-4 +T = 0.25 +10-7 +C +10-10 +10-13 +10-16 +50 +100 +150 +200 +250 +300 +Total integration time6 +Conclusion +Transition to turbulence is a long-standing problem in fluid dynamics, for which adopting a dy- +namical system point of view has greatly increased our understanding. Specialized codes such as +ChannelFlow [111, 112], OpenPipeFlow [273] or Semtex [33] are equipped with the set of tools nec- +essary to probe the phase space of the Navier-Stokes equations. Yet, by design, these are limited to +canonical configurations (e.g. plane Poiseuille and plane Couette flow for ChannelFlow) with infinite +spans, and no such library is available for a general-purpose CFD code. This work therefore aims +at providing a general-purpose introduction to the Krylov methods underlying numerous recent +works on stability analysis of really high-dimensional systems. These methods are implemented in +nekStab, an open-source and user-friendly toolbox to perform large-scale bifurcation analysis in +Nek5000. +Using a time-stepper formulation and leveraging Krylov-based techniques, nekStab is capable +of computing fixed points and periodic orbits, as well as computing the leading eigenpairs and +singular triplets of the corresponding linearized Navier-Stokes operator to characterize their stability +properties. The capabilities of nekStab and its underlying Krylov methods are showcased in a +number of test cases available in the literature, including the canonical cylinder flow [19], the +annular thermosyphon [157], the harmonically forced jet [238] and the flip-flop instability in the +wake of side-by-side cylinders [59]. In all cases, excellent agreement has been obtained with the +results available in the literature. +We believe that this bifurcation analysis plug-in for a code as established as Nek5000 could have +far-reaching applications, both in academia and industry, and could improve our understanding of +the transition to turbulence in non-canonical configurations. By making the code open source, we +hope to foster the ideas and efforts of our community as a whole and incorporate them into the +development of a high-quality tool from which we can all benefit. +Data availability +The code and scripts presented in this review are freely available at https://github.com/nekStab. +Acknowledgment +This work is built on the research of numerous former Ph.D. students, including Fr´ed´eric Alizard [4], +Stefania Cherubini [64], Jean-Christophe Loiseau [153], Alessandro Bucci [47], Mirko Farano [96], +Francesco Picella [204] and Ricardo A. S. Frantz [103]. The authors acknowledge the support of +GENCI under the following projects: A0072A06362/2020, A0092A06362/2021, and A0112A06362/2022. +We are also very grateful to Laurette Tuckerman for all the valuable discussions and insights that +greatly improved this manuscript. +A +nekStab, an open source toolbox for Nek5000 +The numerical methods introduced in this paper are implemented and validated in a toolbox for +linear stability analysis of steady and periodic flows. nekStab is an open-source toolbox that im- +plements all the algorithms described in this article in less than 9000 lines of Fortran 90. The +toolbox uses the flexible and highly parallel data structure of Nek5000, which allows efficient com- +putation of flows in complex geometries. In the current implementation, when compiling Nek5000 +nekStab is appended as a submodule so that the core of Nek5000 is only minimally modified. +47 + +Nek5000, the workhorse of nekStab, is valued for its minimal dependencies and parallel perfor- +mance, enabling computations on simple laptops up to high-performance computers. Great care +has been taken to limit the additional dependencies required to use nekStab, thanks to a non- +invasive and self-contained design, no additional dependencies are introduced, and the need to +modify the source code of Nek5000 has also been eliminated. To avoid compatibility issues and +ease of use, nekStab interacts only with Nek5000 through the existing user interface, without re- +quiring any changes to the original source code while retaining its native performance. nekStab +directly uses Nek5000 time-stepper and closely adheres to its data structure, but only interacts with +it through the existing user interface. Given the design strategy for nekStab, potential conflicts +with future updates to the main solver are unlikely, as the variables are separated by subroutines +crafted to interface with Nek5000 variables. Like Nek5000, our toolbox also makes use of a few +subroutines from the Linear Algebra PACKage (LAPACK) provided with nekStab to avoid exter- +nal dependencies (version 3.10 available at github.com/Reference-LAPACK included). All results +presented in this work were computed with nekStab using the latest version of Nek5000 (commit +7ae03b1), available in github.com/Nek5000. The well-commented source code is maintained online +in the public Git repository at github.com/nekStab, as is the collaborative online documentation +at nekstab.github.io with instructions and lightweight examples that can be quickly computed +on a laptop. These include canonical and more complex flows, such as flow past a cylinder or ad- +jacent cylinders, a time-periodic axisymmetric jet, open and closed cavity flow, a stratified annular +thermosyphon, flow past an airfoil, a backward step, sudden channel expansion, channel flow, and +a Blasius boundary layer. The examples are compared to a reference case from the literature and +aim to provide all the elements that new users might need to develop their own cases. +To the best of our knowledge, nekStab is the first general-purpose computational framework +capable of stabilizing fully 3D unstable periodic orbits (forced or unforced) and fixed points, as well +as computing direct, adjoint modes and transient growth analyses for both steady and periodic +flows. The menu of options includes a matrix-free Newton GMRES solver as well as other classical +techniques such as selective frequency damping (SFD) [1], BoostConv [70], Time-Delayed Feedback +(TDF) [210, 237], and Dynamic Mode Tracking (DMT) [211], as well as post-processing routines +such as the kinetic energy budget of the leading modes based on the Reynolds-Orr decomposi- +tion [38, 154], steady-state base flow, and sensitivity analyzes [174]. +To facilitate adoption by the community (both academic and industrial), nekStab is released +under the permissive BSD 3-Clause license. All source code, examples, scripts, and tutorials can +be viewed and downloaded for free at github.com/nekStab or nekstab.github.io. +We foresee more updates and future developments for nekStab on the horizon: +• integration of the algorithms proposed in [235] for the synthesis of linear optimal LQR con- +trollers for large-scale systems, +• integration of the optimization algorithms in [102] and already implemented in Nek5000 by +M. Farano [97, 98] for linear and nonlinear optimal perturbation analysis, +• extension of the Newton-Krylov solver with pseudo-arclength continuation to compute branches +of solutions even in the presence of folding points, +• Computation of Lyapunov exponents using the algorithms presented in [27] and [104], +48 + +B +Proof of equation (57) +Let us prove that, given the Arnoldi factorization +AQ = QH + βeT +k qk+1, +the approximation of the ith eigenvector vi in the Krylov basis Q satisfies +���� +Ä +I − QQT ä +vi +���� ≤ +Ü +n +� +j=1 +j̸=i +|αj| +|αi| +ê +ε(k) +i +. +(58) +Recall furthermore that the vector b used to seed this Krylov vector can be expressed as a linear +combination of the eigenvectors, i.e. +b = +n +� +i=1 +αivi. +(59) +The proof below follows closely the derivation given by Elias Jarlebring in his lecture notes at KTH +(see https://www.math.kth.se/na/SF2524/matber15/arnoldiconv.pdf). It proceeds in three +steps. +Step 1 +Consider an arbitrary vector u ∈ Cn. Then the problem +minimize +z∈Ck +∥u − Qz∥2 +(60) +is a least-squares minimization problem. +The matrix Q being orthonormal, its Moore-Penrose +pseudoinverse Q† is equal to QT . Hence, the vector z solution to the minimization problem is +simply given by z = QT u. It implies in particular that +min +z∈Ck ∥u − Qz∥2 = ∥ +Ä +I − QQT ä +u∥2. +(61) +Step 2 +Our objective is to obtain an upper bound for the term ∥ �I − QQT � vi∥2, where vi is the +ith eigenvector of A. The proof is simplified by rescaling the right-hand side with αi, i.e. +∥ +Ä +I − QQT ä +αivi∥2 = min +z∈Ck ∥αivi − Qz∥2 += +min +y∈Kk(A,b) ∥αivi − y∥2. +(62) +The Krylov subspace Kk(A, b) can be characterized with polynomials. The statement y ∈ Kk(A, b) +is thus equivalent to the existence of a polynomial p ∈ Pk−1 such that y = p(A)b. Hence +∥ +Ä +I − QQT ä +αivi∥2 = min +p∈Pk−1 ∥αivi − p(A)b∥2. +(63) +Step 3 +The last step consists in inserting the expansion of b in terms of the eigenvectors. This +leads to +∥ +Ä +I − QQT ä +αivi∥2 = min +p∈Pk−1 ∥αivi − p(A) +n +� +j=1 +αjvj∥2 += min +p∈Pk−1 ∥αivi − +n +� +j=1 +αjp(µj)vj∥2. +(64) +49 + +It can then easily be shown that the expression above is bounded from above by +∥ +Ä +I − QQT ä +αivi∥2 ≤ +Ü +n +� +j=1 +j̸=i +|αj| +ê +· ϵ(k) +i +, +(65) +where ϵ(k) +i +is given by +ϵ(k) +i += +min +p∈Pk−1 +p(µi)=1 +max +j̸=i (|p(µj)|). +(66) +The proof of equation (57) is completed by dividing this upper bound by |αi|. +References +[1] E. ˚Akervik, L. Brandt, D. S. Henningson, J. Hœpffner, O. Marxen, and P. Schlatter. Steady +solutions of the navier-stokes equations by selective frequency damping. Phys. Fluids, 18(6): +068102, 2006. +[2] S. Albensoeder, H. C. Kuhlmann, and H. J. Rath. Three-dimensional centrifugal-flow insta- +bilities in the lid-driven-cavity problem. Phys. 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Fluids, 7(4):779–794, 1995. +[280] Oleg Zikanov, Dmitry Krasnov, Thomas Boeck, Andre Thess, and Maurice Rossi. Laminar- +turbulent transition in magnetohydrodynamic duct, pipe, and channel flows. Appl Mech Rev, +66(3), 2014. +67 + diff --git a/M9FOT4oBgHgl3EQf1zQJ/content/tmp_files/load_file.txt b/M9FOT4oBgHgl3EQf1zQJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..751fc3ccd2369f5d69c982c71cd4d84e2b29fc95 --- /dev/null +++ b/M9FOT4oBgHgl3EQf1zQJ/content/tmp_files/load_file.txt @@ -0,0 +1,2683 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf,len=2682 +page_content='Krylov methods for large-scale dynamical systems: Application in fluid dynamics Ricardo S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Frantz1, Jean-Christophe Loiseau1, and Jean-Christophe Robinet1 1Arts et M´etiers Institute of Technology, CNAM, DynFluid, HESAM Universit´e, F-75013 Paris, France January 31, 2023 Abstract In fluid dynamics, predicting and characterizing bifurcations, from the onset of unsteadi- ness to the transition to turbulence, is of critical importance for both academic and industrial applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Different tools from dynamical systems theory can be used for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In this review, we present a concise theoretical and numerical framework focusing on practical aspects of the computation and stability analyses of steady and time-periodic solutions, with emphasis on high-dimensional systems such as those arising from the spatial discretization of the Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Using a matrix-free approach based on Krylov methods, we ex- tend the capabilities of the open-source high-performance spectral element-based time-stepper Nek5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The numerical methods discussed are implemented in nekStab, an open-source and user-friendly add-on toolbox dedicated to the study of stability properties of flows in complex three-dimensional geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The performance and accuracy of the methods are illustrated and examined using standard benchmarks from the fluid mechanics literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Thanks to its flexibility and domain-agnostic nature, the methodology presented in this work can be applied to develop similar toolboxes for other solvers, most importantly outside the field of fluid mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 1 Introduction The transition to turbulence is a long-standing problem in fluid dynamics, pioneered by Osborne Reynolds [217].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' As mathematical tools have developed, a dynamical systems point of view has led to a better understanding of this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Before the advent of computers, theoretical analyses had to rely on simplifying assumptions, the most important ones being the parallel flow assumption and that of infinitesimal perturbations forming what we now know today as local stability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For simple shear flows, these assumptions lead to the famous Orr-Sommerfeld-Squire equations ïÅ ∂ ∂t ã ∇2 − U′′ ∂ ∂x − 1 Re∇4 ò v = 0, ï ∂ ∂t + U ∂ ∂x − 1 Re∇2 ò η = −U′ ∂v ∂z , (1) where U(y) is the base flow velocity profile, v(x, y, z, t) is the wall-normal velocity component of the perturbation, and η(x, y, z, t) its wall-normal vorticity component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Despite their simplicity, these assumptions led to important theorems in hydrodynamic stability theory: Rayleigh’s in- flection point criterion [214], Fjørtoft [101] or Squire’s theorem [243].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' They also led to a better 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='12940v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='flu-dyn] 30 Jan 2023 understanding of non-normality and to the development of nonmodal stability analysis [215, 231].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Although U(y) is formally a steady solution of the Navier-Stokes equations, a tremendous amount of understanding has been gained by replacing it with simple approximations such as the Bachelor vortex sheet model or the piecewise linear approximation of the Blasius boundary layer profile in the seminal work of Tollmien and Schlichting [229, 256].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Using a normal mode ansatz, the velocity fluctuation can be decomposed using a Fourier expansion v(x, y, t) = ˆv(y) exp (i(αx − ωt)) , (2) and similarly for the vorticity fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Determining the stability of the system then amounts to solving a generalized eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Depending on the assumptions about the wavenumber α and the frequency ω, these stability analyses fall into two categories Temporal stability: defined by α ∈ R and ω ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In this context, one aims to determine whether a fluctuation grows over time at a given streamwise position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Spatial stability: defined by α ∈ C and ω ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In this context, one investigates whether forcing at a particular frequency causes the perturbation to grow in space while being advected by the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' An important milestone was achieved by Huerre & Monkewitz [127, 128] by letting both α and ω be complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Depending on subtle properties of the dispersion relation in the complex plane, they introduced the dichotomy between absolute and convective instabilities, thus establishing the first connection between the local stability properties of the flow and its spatiotemporal evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' An instability is classified as absolute if the perturbation grows in place (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' its group velocity is zero), otherwise as convective if it grows as it propagates before leaving the region of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For more details, see [73, 127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This connection between local and global properties of flow was refined by Monkewitz, Huerre and Chomaz [185] using the WKBJ formalism and led to the weakly non-parallel flow assumption which is, however, still very limiting and hardly applicable to flow configurations of practical interest where separation is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For a good numerical overview of local stability theory, interested readers are referred to the book by Schmid & Henningson [232].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This leap in understanding the nature of instabilities allowed for better explanations and ex- panded the limiting dichotomy of open and closed flows, as the dynamics themselves could now be categorized as noise amplifiers or oscillators (more details given in [73, 128]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This distinction is of great importance, for example, for the selection and design of flow control strategies or the placement of sensors and actuators (see the review by Schmid and Sipp [233] and also Cossu [74]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Flow configurations of a noise-amplifying nature are much more difficult to control and predict because the dynamics are sensitive to both the amplitude and the spectral content of the incoming disturbance (viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' vibrations, acoustics, or turbulence intensity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In such flows, the incoming dis- turbances can excite otherwise stable modes and begin to extract energy from the base flow while they are transported downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Natural oscillators, on the other hand, are characterized by the presence of a dominant unstable structure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' a global instability), which locates the physical mechanism that extracts energy from the base flow in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' When the instability is suppressed, the flow becomes stable and returns to the laminar state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' More complex flow configurations (such as the jet in crossflow) may exhibit convective noise amplifier behavior or self-sustaining oscillatory behavior, depending on the combination of control parameters [178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Problems such as the one presented in equation (1) can be analyzed theoretically, but in practice, it is common to discretize the resulting equations in the wall-normal direction using spectral meth- ods such as Chebyshev polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A famous example is the work of Orszag [195] on the temporal 2 stability of the canonical plane Poiseuille flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' At about the same time as [65, 127, 128, 185], com- puters and numerical methods began to reach sufficient maturity that the (weakly non-)parallel flow assumption could be relaxed and the spectral decomposition of the Navier-Stokes operator linearized in the vicinity of a truly two-dimensional base flow started to be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In fact, the computation of 2D baseflows in hydrodynamics began in the 1970s with direct methods [29, 182, 234, 275] before other, more efficient numerical approaches became popular and years later enabled the computation of 3D baseflows with increasing computer power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In the mid-1980s, Jackson [131] and Zebib [278] obtained 2D steady states and computed stabil- ity analysis on the flow past a circular cylinder using full-matrices (Jackson used iterative methods to approximate the leading eigenvalues).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' At the same time, the first use of a time-dependent numerical solver and Krylov methods for fluid dynamics seems to point to the work of Erikson & Rizzi [93], which coincides with the eigenvalue calculations of Tuckerman [262] and Marcus & Tuckerman [171, 172] on the flow between concentric rotating spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Similar numerical methods have been explored by Goldhirsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' [113], Christodoulou & Scriven [66], Tuckerman [260], and Edwards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In these works, the prohibitive memory requirements of a matrix-forming approach are replaced by methods that use more accessible computational resources and are now commonly referred to as the matrix-free/time-stepping approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These advances in algorithms eventually enabled the computation of 3D eigenmodes evolving on 2D solutions in the 1990s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This began with the work of Natarajan & Acrivos [188] and Ramanan & Homsy [212] on the lid-driven cavity flow and later the analysis of Barkley & Tuckerman [22] on the perturbed plane Couette flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' To distinguish these analyses, which solve the linearized 2D equations, from a local stability framework relying on a (weakly-) parallel flow approximation, Theofilis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' [250] has called them (bi-)global stability analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Since then, the linear stability of numerous two-dimensional flow configurations have been studied: the lid-driven cavity flow [2, 251], the backward-facing step [146], or the two-dimensional flow past a bump [90, 91], to name just a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Because of its importance for the development of local stability theory, the two-dimensional boundary layer flow has also been the focus of many investigations [5, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Although matrix-free methods were already available, it is important to mention that several of the previously mentioned papers still considered the explicit construction of the linearized Navier- Stokes operator in combination with standard algebraic solvers to compute its eigenpairs (other examples include [106, 118, 251]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For a comprehensive review of research contributions up to the early 2010s, with particular emphasis on this matrix-forming approach, see Theofilis [252] and Juniper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' [132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Yet, over the past decade, the time-stepping framework has become increasingly popular and enabled the investigation of the stability properties of fully three-dimensional flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A large body of works has focused on two configurations, namely the jet in crossflow [13, 129, 203] or the boundary layer flow past three-dimensional roughness elements [43, 46, 48, 67, 145, 154, 162, 276].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The stability of lid-driven and shear-driven three-dimensional cavities with spanwise end-walls has also been investigated in [99, 107, 143, 151, 155, 205].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The same methodology has also been employed to compute the leading optimal perturbation [231] in magnetohydrodynamic flows [280], or best exemplified by [35, 36] on backward facing steps and stenotic pipe flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' It has also been used to solve high-dimensional Ricatti equations for linear optimal control in [235], or to study the stability properties of flow governed by the compressible Navier-Stokes equations with or without shocks [94, 95, 226].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These include modal and non-modal stability of compressible boundary layers [49, 50, 117, 125, 218], cavities [41, 247, 253, 277], wavepackets in jets [26, 192, 236], transonic buffet [75– 77, 199, 200, 255], including the flow past the NASA Common Research wing Model [254, 255], wakes [181] and bluff bodies [163, 164, 226, 227].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 3 The matrix-free approach has been the key to efficient computation and (Floquet) stability analysis of time-periodic solutions, beginning with Schatz, Barkley & Swinney [228], followed by the secondary instability of the flow past a circular cylinder with the canonical work of Barkley & Henderson [19] and later by the study of Blackburn, Marques & Lopez [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Other work includes the backward-facing step flow [24], the study of the stability properties of a pulsatile stenotic pipe flow [36], the flip-flop instability in the wake of two side-by-side cylinders [59], the secondary bi- furcation in a shear-driven cavity flow [28], or the study of the vortex pairing mechanism in a harmonically forced axisymmetric jet [238].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In parallel with these developments in the hydrody- namic stability community, similar numerical methods and tools have been explored in the com- munity looking at turbulence from the point of view of a dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Matrix-free methods and clever exploration of flow symmetries can significantly reduce computational costs and allow the calculation of exact coherent states1 in the turbulent basin of attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These exact coher- ent states include relative periodic orbits [141, 213], chaotic saddles [136, 201, 270, 274] or edge states [56, 130, 241].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Interested readers may refer to the specialized codes Channelflow [111] or Openpipeflow [273] and the corpus of related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' It is also noteworthy that in the special case of weakly non-parallel shear flows, the development of parabolized stability equations [123, 149] paved the way for the estimation of the neutral stability curves of the Blasius boundary layer [31, 121, 152] and its secondary instabilities [122, 124], which may involve curvature effects inducing centripetal G¨ortler vortices [166, 216].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Building on the framework of Orszag and aiming at the geometric flexibility of the finite ele- ment method, Patera [202] and Maday & Patera [165] laid the foundation of the spectral element method (SEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' SEM has since become a popular discretization strategy in computational fluid dy- namics (CFD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Within the incompressible hydrodynamic stability community, the spectral element solver Nek5000 [100] has established itself as one of the leading high-performance open-source CFD codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Most of the aforementioned three-dimensional stability analyses heavily relied on Nek5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Except for the KTH Framework2, relatively few toolboxes have been developed for Nek5000 de- spite its large user base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Even then, the capabilities of this toolbox (as far as linear stability is concerned) are limited to simple fixed-point calculations using the selective frequency damping ap- proach [1], while the leading eigenpairs of the linearized Navier-Stokes operator are calculated using PARPACK [148, 176], at the expense of introducing new dependencies for the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Linear stability analysis capabilities are also available to Nek5000’s brethren, including Nektar++ [58, 135], which can handle a variety of mesh types, and Semtex [33] designed for spanwise periodic flows, with examples given by [3, 32, 92, 170, 239].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The finite element code FreeFem++[120] can also be used to extract the Jacobian matrix directly with examples including [68, 69, 133, 173, 174, 277].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The solver capabilities can be extended using the StabFem [95] MATLAB suite for compressible flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Finally, the FEniCS [6] project is also another consolidated alternative with a Python interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Re- cently several other spectral-based Python projects for solving partial differential equations became available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These include SpectralDNS [186, 187], FluidSim [10, 184], Dedalus [51] and Coral [183].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The finite volume framework BROADCAST [209] has become available for the study of 2D curvilinear structured grids with support for nonlinear and linear calculations of compressible flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The aim of the present work is by no means to be an exhaustive review, but rather a compre- hensive introduction to the Krylov methods underlying the most recent works on the stability of 1Exact coherent states can be fixed points in the original reference frame or in a co-moving reference frame (in which case they are called traveling waves) or true periodic states [82, 261].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 2Freely available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='com/KTH-Nek5000/KTH_Framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Apart from some linear stability capabil- ities, this toolbox also provides additional capabilities such as full restarts, turbulence statistics, additional boundary conditions, or incorporating user variables and various volume forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 4 very large-scale dynamical systems such as fully three-dimensional flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For that purpose, this manuscript is organized as follows: first, brief overviews of the theoretical framework and numerical methods are given in section 2 and section 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A collection of examples illustrating the use of these techniques in fluid dynamics is provided in section 4, while some particular theoretical and practical points are discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Finally, conclusions and perspectives are given in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 2 Theoretical framework We focus on the analysis of the stability properties of high-dimensional nonlinear dynamical systems, typically arising from the discretization of partial differential equations such as the incompressible Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Once discretized, the governing equations are generally expressed as a nonlinear system of first-order ordinary differential equations dXj dt = Fj ({Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' i = 1, · · · , n} , t) , where n is the dimension of the discretized system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Using the notation X and F for the sets {Xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' j = 1, · · · , n} and {Fj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' j = 1, · · · , n}, this system can be written as dX dt = F(X, t), (3) where X ∈ Rn is the state vector of the system and t is a continuous variable that denotes time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' If the system is supplemented with constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' the divergence-free constraint in incompressible fluid dynamics), F is then understood as the restriction of dynamics in the feasible set of such constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' One can also consider the equivalent discrete-time system Xk+1 = Φτ(Xk), (4) where Φτ(X) is the forward map defined as Φτ(X) = � τ 0 F(X(t), t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (5) Such a discrete-time system may result from the time discretization of the governing equations (with τ the sampling period) or in the study of periodic orbits (with τ the period of the orbit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In practice (and due to discretization errors), an approximation to the action of an operator is made by computing many (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' τ/∆t) small time-steps (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' ∆t ≪ 1), each consisting of a rational or polynomial approximation to the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The term timestepper, coined in 2000 by Tuckerman & Barkley [264], refers to the adaptation of a time integration code to perform bifurcation analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In what follows, we consider an exact operator notation as a shorthand notation for the timestepper approximation of operators (linear or nonlinear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In the following sections, we present a definition of fixed points, periodic orbits, and linear stability analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These are the fundamental concepts required to characterize the properties of the system under investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In particular, we focus on the modal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' asymptotic) and non- modal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' finite-time) stability analysis, which are the classical and a more modern approach that has become increasingly popular in fluid dynamics in recent decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' We note that part of the community has also shifted its attention to nonlinear optimal pertur- bations, which is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Interested readers are referred to the work of [139] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 Fixed points and periodic orbits Nonlinear dynamical systems, such as equation (3) can admit different solutions or attractors that form the backbone of their phase space: fixed points (steady dynamics), periodic orbits (periodic dynamics), tori (quasi-periodic dynamics) or strange attractors (chaotic dynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Hereafter, our attention will be solely focused on fixed points and periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 Fixed points For a continuous-time dynamical system described by equation (3), the fixed points X∗ are partic- ular equilibrium solutions satisfying F(X∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (6) Similarly, for a discrete-time system described by equation (4), fixed points are solutions to Φτ(X∗) = X∗ ∀τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (7) These particular solutions are thus characterized by the absence of dynamics: the system is in a steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Since we are dealing with nonlinear equations, both equation (3) and equation (4) can have a multitude of fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This is illustrated by a dynamical system as simple as the Duffing oscillator � � � ˙x = y, ˙y = −1 2y + x − x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (8) Because of the stabilizing cubic term in the y-equation, the Duffing oscillator admits three fixed points a saddle at the origin X∗ = (0, 0), two linearly stable spirals located at X∗ = (±1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These different fixed points, along with typical trajectories, are depicted in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A multiplicity of fixed points also occurs in nonlinear dynamical systems as complex as the Navier-Stokes equations, see for instance [116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The decision of which of these fixed points is the relevant one from a physical point of view depends on the problem and is left to the judgment of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The computation of equilibria is a cornerstone for all analyses described in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Numerical methods to solve equation (6) and equation (7) are discussed further in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2 Periodic orbits The second type of equilibria of interest to us is periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Such solutions are characterized by dynamics repeating themselves after a given period τ ∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' X(t + τ ∗) = X(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (9) They can also be understood as fixed points of the forward map Φτ for τ = τ ∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' X∗ = Φτ(X∗) for τ = τ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (10) In practice, τ ∗ is often unknown, and therefore one must solve simultaneously for a point X∗ on the orbit and the period τ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Moreover, any point on the orbit satisfies the equation above so that equation (10) admits an infinite number of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' To close the system, a phase condition often 6 Figure 1: Phase portrait of the unforced Duffing oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The red dots denote the three fixed points admitted by the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The blue (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' orange) thick line depicts the stable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' unstable) manifold of the saddle point located at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Grey lines highlight a few trajectories exhibited for different initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' needs to be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A canonical example of a periodic orbit in fluid dynamics is the periodic vortex shedding in the wake of a two-dimensional cylinder at a low Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' As for fixed points, a nonlinear dynamical system may admit multiple periodic orbits, each with its own period τ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This is particularly true when the system evolves on a strange attractor (chaotic dynamics) on which an infinite number of unstable periodic orbits (UPO) coexist with arbitrary periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These findings go back to Kawahara & Kida [136], who pioneered the extraction of periodic orbits in a fully three-dimensional plane Couette flow using a Newton root-finding technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The discovery of a large number of UPOs buried in turbulent attractors for various flow configurations supports the view that turbulence is a very high-dimensional dynamical system whose trajectories repeatedly visit unstable exact coherent structures [87, 137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Since the discovery of recurrent spatiotemporal structures concealed in a turbulent attractor, UPOs have proven to be favorable kernels [83] for predicting turbulence statistics due to their harmonic temporal structure [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These invariant solutions can be naturally sustained by the flow and have been found to be energetically relevant for predicting or even reconstructing turbulence statistics (if enough of them are found) [62, 159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Despite growing interest since the first UPOs were obtained, the methods used to obtain them involve brute force, although attempts are being made to change this [197].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Considering the R¨ossler system [219] � � � � � ˙x = −y − z, ˙y = x + ay, ˙z = b + z(x − c), (11) for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1, b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1, and c = 14, figure 2 shows its famous strange attractor as well as two UPOs embedded in this attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These were obtained using a simple shooting Newton method and continuation initialized with their stable counterparts at lower values of the control parameter c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 7 2 O 2 2 1 0 1 2 XFigure 2: Strange attractor for the R¨ossler system with parameters a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1, b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 and c = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Colored lines depict a τ1 (red) and τ2 (blue) unstable periodic orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2 Modal and non-modal linear stability To avoid repetition, we restrict ourselves in this section to the modal and non-modal stability of fixed points, although these concepts can easily be applied to periodic orbits by replacing the Jacobian matrix with the monodromy matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The linearly stable or unstable nature of the fixed point X∗ is characterized by the fate of infinitesimal perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' If perturbations eventually decay, the equilibrium is considered stable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' otherwise, it is considered unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Note, however, that an infinite time horizon is allowed for the return to equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Thus, a fixed point can be classified as stable even if a small perturbation transiently departs very far from it before returning toward it only asymptotically with t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This distinction between asymptotic and finite-time evolution gives rise to the concepts of modal and non-modal stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The reader will find a detailed discussion of these concepts in fluid dynamics in [230, 231].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The dynamics of a perturbation x = X − X∗ is governed by ˙x = F (X∗ + x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (12) Assuming x is infinitesimally small, F(X) can be approximated by its first-order Taylor expansion around X = X∗, leading to ˙x = Lx, (13) where L = ∂F ∂X, (14) is the n × n Jacobian of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Starting from an initial condition x0, the perturbation at time τ is given by x(τ) = exp(τL)x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (15) 8 UPO1 UPO2 Z xThe operator Mτ = exp(τL) is the exponential propagator of the linearized system and corresponds to the Jacobian of the forward map Φτ linearized in the vicinity of the fixed point X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For periodic orbits, this linearized propagator is defined as Mτ = � τ 0 L(t) dt, (16) with L(t + τ) = L(t) and is known as the monodromy matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' We again use the notation of the exact operator as shorthand for its discrete counterpart obtained by integration over many small timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' System operators can be linearized exactly by using a timestepper code to solve the set of linearized equations derived analytically, or by using automatic differentiation tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Such an exact linear operator is in itself a discrete approximation, again requiring temporal integration over many small timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Alternatively, the action of the linearized operator can be emulated using less accurate finite-difference approximation [26, 177].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' While the computational cost remains the same for a first-order approximation, it increases with the order of the finite-difference scheme considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 Modal stability Introducing the (Euclidean) 2-norm of x(τ) and the eigendecomposition L = VΛV−1, (17) one can easily show exp(2τλr) ≤ ∥ exp(τL)∥2 2 ≤ κ(V) exp(2τλr), (18) where λr = Re(λ1) is the real part of the leading eigenvalue (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' with greatest real part) of L and κ(V) = ∥V∥2∥V−1∥2 is the condition number of the matrix of eigenvectors V (with κ ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Asymptotic stability is characterized by lim τ→∞ ∥ exp(τL)x0∥2 = 0 ∀ x0, (19) so a sufficient condition is that all eigenvalues of L have a negative real part (equivalent to all eigenvalues of Mτ being inside the unit circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The perturbation decaying the slowest is given by the eigenvector v1 associated with the least stable eigenvalue λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Hereafter, a fixed point X∗ is classified as follows if Re(λ1) > 0, the dynamics of simulations initialized with an initial condition non-orthogonal to the leading eigenvector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' x0 ̸⊥ v1) will grow exponentially rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The fixed point X∗ is deemed linearly unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' if Re(λ1) < 0, the dynamics of simulations initialized with any initial condition will eventually decay exponentially fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The fixed point X∗ is thus linearly stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The case Re(λ1) = 0 is special and corresponds to a non-hyperbolic fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Its stability cannot be determined by the eigenvalues of L alone and one has to resort to weakly nonlinear analysis or center manifold reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Interested readers can refer to [60, 167, 240, 271] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2 Non-modal stability The upper bound in equation (18) involves the condition number κ(V) of the matrix of eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This leads to the classification of linear systems into two distinct sets with fundamentally different finite-time stability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Systems for which κ(V) = 1 are called normal operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In this case, the eigenvectors of L form an orthonormal set such that V −1 = VH, (20) where superscript H denotes the Hermitian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' complex-conjugate transpose operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The finite- time and asymptotic stability properties of the system are identical, and the dynamics cannot exhibit transient growth: analyzing the spectrum of L is sufficient to fully characterize the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' When κ(V) > 1, the matrix L is said to be non-normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This can be defined by introducing the adjoint operator L† satisfying ⟨y|Lx⟩ = ⟨L†y|x⟩, (21) where ⟨·|·⟩ is a suitable inner product (typically the inner product induced by the ℓ2 norm), along with appropriate boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Non-normality then corresponds to the fact that L and its adjoint L† do not commute, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' L†L ̸= LL†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (22) Its eigenvectors no longer form an orthonormal basis for Rn and the dynamics may exhibit transient growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' From a physical point of view, transient growth can be understood as a constructive interference involving almost colinear eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The larger the non-normality of L, the larger the maximum transient growth with perturbations being (possibly) amplified by several orders of magnitude before the exponential decay eventually takes over (assuming all the eigenvalues have negative real parts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In the most extreme scenario, this non-normality is characterized by L admitting a non-diagonalizable Jordan block leading to algebraic growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' More details on adjoint operators are provided in [126, 160, 257].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Given this observation, one can now ask a more subtle question about the stability of the fixed point X∗, namely How far from the fixed point X∗ can an arbitrary perturbation x0 go (or equivalently to what extent can it be amplified) at a finite time τ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The answer to this question can be obtained by solving the following optimization problem G(τ) = max x0 ∥ exp(τL)x0∥2 2 ∥x0∥2 2 , (23) where ∥ exp(τL)∥2 2 is the vector-induced matrix norm optimizing over all possible initial conditions x0 and G(τ) is the maximal amplification gain of the perturbation at time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Introducing the singular value decomposition of the exponential propagator Mτ Mτ = UΣVH, (24) the maximum gain is simply given by G(τ) = σ2 1, (25) where σ1 is the largest singular value of Mτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The optimal initial condition x0 is then given by the first right singular vector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' x0 = v1) while the associated response is x(τ) = σ1u1, where u1 is the leading left singular vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' When/if one has access to the adjoint operator, computing 10 these quantities can also be recast as an eigenvalue problem (EVP) rather than a singular value decomposition (SVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In fluid dynamics, this concept of non-normality and optimal perturbations leads to a better understanding of the formation and ubiquity of velocity streaks in the transition to turbulence of wall-bounded shear flows [38–40, 258].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' It also sheds some light on the importance of shear layer instability [25, 35, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Extension to periodic orbits has been considered for instance in [36, 170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Although not considered herein, a similar concept exists in the frequency domain, leading to a resolvent analysis (see [231] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Illustration Let us illustrate the concepts of optimal perturbation and transient growth using a simple flow configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For this purpose, consider the incompressible flow of a Newtonian fluid induced by two flat plates moving in opposite directions in the plane, as sketched in figure 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The resulting flow, known as plane Couette flow, is given by U(y) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' It is a linearly stable fixed point of the Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Despite its linear stability, subcrit- ical transition to turbulence due to finite amplitude perturbations can occur at Reynolds numbers as low as Re = 325 [168].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Without delving too much into the mathematical and physical details of such a subcritical transition, part of the explanation can be given by linear optimal pertur- bation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The dynamics of an infinitesimal perturbation x = �v η�T , characterized by a certain wavenumber k = αex + βez, is governed by the Orr-Sommerfeld-Squire equations (matrix counterpart of equation (1)) written as d dt ïv η ò = ïLOS 0 C LS ò ïv η ò , (26) with v the wall-normal velocity component of the perturbation and η its wall-normal vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' LOS denotes the Orr-Sommerfeld operator, while LS and C represent the Squire operator and the cou- pling term, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For certain pairs of wavenumbers, this Orr-Sommerfeld-Squire operator is highly non-normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In particular, perturbations with non-zero spanwise wavenumbers can experi- ence large transient growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This is illustrated in figure 3(b), where the evolution of the optimal gain G(τ) is shown for different spanwise wavenumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The maximum amplification over all target times and wavenumber pairs is Gopt ≃ 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The initial perturbation x0 is shown in figure 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This perturbation corresponds to streamwise-oriented vortices that eventually lead to streamwise velocity streaks, as shown in figure 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Although this perturbation eventually decays exponen- tially fast in a purely linear framework, it has been shown that its transient amplification even at moderately small amplitude can be sufficient to trigger the transition to turbulence when used as an initial condition in a nonlinear simulation [160].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For more details on subcritical transitions and the extension of optimal perturbation analysis to nonlinear operators, interested readers may refer to [231] and [139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='3 Bifurcation analysis The eigenvalue analysis of the Jacobian matrix L or monodromy matrix Mτ plays a key role in determining the type of bifurcation that occurs when a control parameter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' the Reynolds number) is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In the remainder of this section, a brief overview of the standard bifurcations and their correspondence to eigenvalues is given for completeness (see figure 4 for a schematic representation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 11 Figure 3: Illustration of the optimal perturbation analysis for the plane Couette flow at Re = 300 for streamwise wavenumber α = 0: (a) Schematic of the flow, (b) Optimal gain curve for different spanwise wavenumbers β, (c) Optimal perturbation in-plane velocity (v, w) and (d) optimal response out of plane velocity (u) for β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The optimal perturbation consists of streamwise oriented vortices, while the corresponding response at time T consists of high and low-speed streaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Reproduced from [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Figure 4: Eigenvalue patterns associated with the standard bifurcations encountered for fixed points (a and b), and for limit cycles (c to e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In each case, the shaded region indicates the stable part of the spectrum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' the lower complex half-plane for fixed-point stability, and the unit disk for periodic orbits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 12 100 β=2 β=3 h 50 β= 4 0 50 100 150 200 250 T b 1 0 1 2 2 0 2 Z Z d(a) Pitchfork/Saddle-node (b) Hopf 说[入] 况[入] [] (c) Pitchfork/Saddle-node (d) Period-doubling/Flip (e) Neimark-Sacker/Secondary-Hopf s[μ] sμ] [n] [n]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 Bifurcations of fixed points The asymptotic stability properties of fixed points are related to the eigenvalues of the linearized operator L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Bifurcation analysis is concerned with how these properties evolve as the parameters of the system are varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For simplicity, we will only consider situations where a single parameter is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The value of the control parameter at which a change in stability occurs is a bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The bifurcations most commonly encountered in mechanics are the pitchfork, saddle-node, and Hopf bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Each leads to a qualitatively different behavior before and after the bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Pitchfork and Hopf bifurcations come in two flavors, namely subcritical and supercritical (depending on whether the solutions created at the bifurcation point are themselves stable or not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Pitchfork bifurcation This type of bifurcation is often encountered in systems with symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The canonical example in mechanics is that of a flexible beam on top of which a static load is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Below a critical load, the beam remains upright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' As the load increases, the beam suddenly buckles to the left or right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The upright position becomes linearly unstable, and two new stable equilibria are created as the bifurcation point is crossed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Mathematically, the pitchfork bifurcation can be distilled into the following normal form ˙x = λx ± x3, (27) with the sign in front of the cubic term determining whether the bifurcation is super(-) or sub(+) critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' From a linear stability point of view, the linearized operator has a purely real eigenvalue going from being stable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' λ < 0) to unstable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' λ > 0) as we cross the bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This is a necessary, albeit insufficient, condition to conclude that the bifurcation is a pitchfork Moreover, nonlinear analyses (or simulations) are required in order to conclude whether it is supercritical or subcritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Examples from fluid dynamics include the flow in a sudden expansion channel [86, 146], the flow past a sphere [94, 188, 227], the three-dimensional cavity flow [2, 151, 205] or the Rayleigh- B´enard convection between two infinite plates or inside an annular loop [157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Hopf bifurcation The second type of bifurcation commonly encountered is the Andronov-Poincar´e- Hopf bifurcation (or simply Hopf bifurcation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Below the bifurcation point, the system admits a single fixed point and asymptotically stationary dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' As the bifurcation point is crossed, the fixed point changes stability, and a limit cycle associated with periodic dynamics is created in its vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The Hopf bifurcation can be distilled into the following normal form [82], written as ® ˙r = σr ± r3, ˙ϕ = ω + αr2, (28) where r is the amplitude of the oscillations, ϕ their phase, and ω the frequency of the oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The linearized system has a complex-conjugate pair of eigenvalues that change from stable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' σ < 0) to unstable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' σ > 0) when crossing the bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The imaginary part (ω) of this complex-conjugate pair of eigenvalues then dictates the oscillation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Nonlinear analysis (or simulation) is required to determine whether it is super- or subcritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Examples from fluid dynamics include the broad class of so-called flow oscillators such as the two-dimensional cylinder flow [11, 18, 19, 108, 169, 180, 193, 194, 240, 272, 278, 279], the lid-driven and shear-driven cavity flows [17, 53, 179, 220, 240, 277], the jet in crossflow [13, 63, 129] or the roughness-induced boundary layer flow [46, 67, 154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2 Bifurcations of periodic orbits The linearization of the time-periodic flow map Φτ(X) in the vicinity of the periodic orbit X∗ leads to the monodromy matrix (sometimes also known as the time-shift operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' As for fixed points, their eigenvalues (known as characteristic or Floquet multipliers) dictate the asymptotic stability of the periodic orbit under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The stability problem describes the development of small- amplitude perturbations during one period of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' If all Floquet multipliers lie inside the unit disk, the orbit is characterized as asymptotically linearly stable, otherwise as unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Bifurcations occur when one of these Floquet multipliers (or a complex-conjugate pair) steps outside the unit circle when the control parameter is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Physically, the moduli of such Floquet multipliers express the orbit rate of expansion (or contraction) per unit of time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' per period of oscillation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' As an aside, in cases where the limit cycle occurs as an autonomous nonlinear oscillation (not forced), the set of Floquet multipliers contains a unit mode tangent to the limit cycle which corresponds to the time derivative of the base flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A limit cycle in which at least one Floquet multiplier is greater than one expands and is therefore called an unstable periodic orbit (UPO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The most common bifurcations in this context are the pitchfork bifurcation, the period-doubling bifurcation (also known as a flip bifurcation) and the Neimark-Sacker bifurcation (see figure 4 for a schematic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Once again, these are associated with qualitatively different evolutions of the dynamics below and above the bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Pitchfork bifurcation As for its fixed point counterpart, the pitchfork bifurcation of periodic orbits is most often encountered in systems with spatial symmetries and comes in two flavors (super- and subcritical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A canonical example of such pitchfork bifurcations in fluid dynamics is the three- dimensionalization of the periodic vortex shedding in the wake of a circular cylinder [19, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Below the critical Reynolds number Rec ≃ 189, the flow is strictly two-dimensional and exhibits the well- known time-periodic von K`arm`an vortex street.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' All Floquet multipliers lie within the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' When the Reynolds number is increased, a Floquet multiplier leaves the unit circle at µ = 1 and a pitchfork bifurcation occurs3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Given the synchronous nature of the pitchfork bifurcation, the frequency of the vortex shedding remains unchanged, but the spatial structure of the vortices is no longer spanwise invariant: the flow becomes three-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' If one denotes by An the amplitude of this three-dimensionalization after n periods, the corresponding normal form is given by An+1 = µAn ± |An|2An, (29) where µ is the associated Floquet multiplier and the sign of the cubic term determines whether the bifurcation is super- or subcritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Note that, as for fixed points, other types of bifurcations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' saddle-node) are associated with a Floquet multiplier exiting the unit circle at µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Period-doubling bifurcation The second type of bifurcation commonly encountered is the period-doubling bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' They are also known as flip or subharmonic bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In this situation, a Floquet multiplier exits the unit circle at µ = −1 as the bifurcation point is crossed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In the supercritical case, the periodic orbit that was stable below the bifurcation point becomes unstable, and a new orbit with twice the period takes its place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Considering a discrete-time system, 3Because of the spanwise-invariance of the unstable periodic orbit, the dominant Floquet multiplier (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' with the largest magnitude) is actually a double eigenvalue with one eigenvector exhibiting a sine dependence in the spanwise- direction while the other exhibits a cosine dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In this case, the bifurcation is formally known as a circle pitchfork bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 14 the most famous example of this period-doubling bifurcation is the logistic map xn+1 = µxn(1 − xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (30) In a continuous-time framework, this bifurcation can be found in the famous R¨ossler system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Fig- ure 2 shows two such orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The one in blue, denoted as UPO1 with a period τ1 ≃ 6, loses its stability via a period-doubling bifurcation at c = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='376 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Above this critical value, the peri- odic orbit (denoted as UPO2 in figure 2) is created with a period τ2 ≃ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This second orbit then loses its stability through another period-doubling bifurcation (at a critical value c = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='771±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='001) and a new orbit with twice the period (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' τ3 ≃ 24) is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Systems presenting period-doubling bifurcations often exhibit a subharmonic cascade to chaos, a universal behavior of dynamical sys- tems put forth by Feigenbaum and others in the late 1970s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In the context of fluid dynamics, such a subharmonic cascade was shown to occur in a confined Rayleigh-B´enard cell as the Rayleigh number is increased in the seminal work of Libchaber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' [150].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' It was also observed by Buzug et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' [52] experimentally in Taylor-Couette flow and numerically in Couette flow [140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In plane Couette flow, a first bifurcation leads to the formation of a spatially evolving periodic orbit (also called relative periodic orbits or traveling waves) at Re = 236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 [161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Above Re ≈ 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='40, the system undergoes a period-doubling cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' At Re = 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='46 this cascade leads to a chaotic attractor with expo- nentially diverging trajectories that sporadically visit the various previously created UPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' More recently, this subharmonic cascade has also been observed numerically in rotating plane Couette flow under certain conditions [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Period-doubling bifurcations are also observed in harmonically forced shear layers and jets where they give rise to vortex pairing, see [238] for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Neimark-Sacker bifurcation The last type of bifurcation we will consider is the Neimark- Sacker bifurcation named after the works of Neimark [190] in 1959 and Sacker [224] in 1964 (for an overview of their work see the book by Arnold [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This bifurcation is the equivalent of the Hopf bifurcation of a fixed point for periodic orbits and is therefore sometimes called a secondary Hopf bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' From the point of view of linear stability, it is associated with a complex conjugate pair of Floquet multipliers leaving the unit disk at an angle that is neither 0 nor π, according to µ = e±iω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' If the new frequency is rationally related to that of the periodic orbit, the dynamics immediately after the bifurcation remains periodic, but with a different period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Rationally related frequencies always form a set of measure zero in the set of possible imaginary parts of the Floquet exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' If the new frequency is irrationally related to that of the periodic orbit, the dynamics becomes quasiperiodic immediately after the bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In this case, the corresponding phase space object changes from a simple periodic orbit (below the bifurcation point) to a torus (above the bifurcation point) [53, see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 13] in which the fundamental frequency and the smaller new frequency map the large and small circles of the torus, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The dynamics following a secondary Hopf bifurcation can be much more complicated, exhibiting phenomena such as frequency locking, high- order synchronization (Arnold’s tongues), and devil’s staircase, which are beyond the scope of this review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Interested readers are referred to the book by Pikovsky, Rosenblum & Kurths [207].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' To help reveal the structure of the attractor, one can plot the intersections of the trajectories with a plane, called a Poincar´e section, which intersects the attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' As the trajectories cross the plane in different positions, a Poincar´e section on a torus-shaped object will continuously cover a circle [30] (provided the temporal signal is long enough).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The Fourier spectrum of a periodic system that becomes quasiperiodic is characterized by the appearance of a new peak at a frequency f2 that is incommensurate with the fundamental frequency f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Due to nonlinear interactions between these two frequencies, other peaks may appear in the spectra, which can be easily explained by the linear combinations |n1f1 ±n2f2|, where n1 and n2 are integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In fluid dynamics, such a bifurcation has 15 been shown to occur in the wake of two side-by-side cylinders [59], with the corresponding instability known as the flip-flop instability, as well as in a two-dimensional shear-driven cavity [53, 147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Beyond quasiperiodic dynamics Over the years, the transition from periodic and quasi- periodic dynamics to chaos has been studied in detail [114, 248].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' When the control parameter is varied, steady, periodic, and quasi-periodic systems can undergo a progressive loss of stability and the appearance of chaos by following characteristic paths known as routes to chaos [7], with the most relevant ones being: The Feigenbaum path: The system undergoes a cascade of period-doubling bifurcations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The Ruelle-Takens-Newhouse route: gradual generation of incommensurable frequencies by a sequence of (Hopf) bifurcations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The Pomeau-Manneville scenario: the existence of an intermittent alternation of regular phases and chaotic bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These characteristic routes are reviewed in Eckmann [88] and can be identified by time-series analysis of physical or numerical experiments based on classical Fourier analysis to more com- plex phase-portrait reconstructions [42], delayed embedding [115, 249] or recurrence analysis [175] techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In 1971, Ruelle and Takens [221] showed that when the nonlinearity (or coupling) of a quasi- periodic system increases, Hopf bifurcations lead to an increase in the dimension of the torus, eventually becoming structurally unstable and collapses into a strange attractor with non-integer fractal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The route known as Ruelle-Takens-Newhouse (RTN) [191, 221] is associated with the direct appearance of chaos after the formation of a T 3 torus (or even T 4 torus in some cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In perhaps even rarer cases, the T 3 torus can undergo a rapid devil’s staircase to chaos [134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Flows undergoing the RTN path include the convergent-divergent channel flow [119], the Taylor-Couette flow [72] or the flow in the highly curved toroidal pipe flow [55], or even higher dimensional tori in T n with n ≥ 3 [114, 150, 196].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The path of intermittency introduced by Pomeau and Manneville [208] in the 1980s involves an alternation between periodic and chaotic dynamics, although all system parameters remain constant and free of noise (Manneville [167]’s book gives a clear overview).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Beyond the bifurcation point, dynamical systems with intermittency exhibit bursts of irregular motion with higher amplitude amidst regular motion with lower amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The duration of the irregular bursts increases with the control parameters until the chaotic dynamics predominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Depending on the value of the Floquet multiplier at the bifurcation point, there are different evolutionary trajectories for the decay of the periods of laminar phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In the theory of intermittent transitions, Floquet stability analysis provides three classifications for intermittency: Type I is associated with a pitchfork bifurcation (the Floquet multiplier crosses the unit circle at +1 at the bifurcation point), type II with a Neimark-Sacker bifurcation (two complex conjugate eigenvalues), and type III with a period-doubling bifurcation (the Floquet multiplier crosses the unit circle at -1 at the bifurcation point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Note that the latter two cases require a subcritical character of the bifurcation for the appearance of intermittency [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 3 Numerical methods Numerous tools exist to study low-dimensional dynamical systems such as the R¨ossler or Lorenz [158] systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These include AUTO [84, 85] in Fortran or pde2path [266, 267] and MatCont [81] in MAT- 16 LAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' PyDSTool [71] offers similar capabilities in Python, while BifurcationKit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='jl [269] is a corre- sponding Julia package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Except for BifurcationKit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='jl, most of them rely on standard numerical linear algebra techniques that do not scale well for very high-dimensional problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Moreover, they do not necessarily integrate easily with parallel programming, which is ubiquitous when simulating discretized partial differential equations such as Navier-Stokes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Thus, extra care may be needed to interface with the particular data structure of the original code (see Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This section provides a brief overview of the standard iterative techniques used to compute fixed points or periodic orbits of very high-dimensional dynamical systems and to study their stability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These techniques rely on Krylov subspaces and associated Krylov decompositions [244] introduced in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The Newton-Krylov method for fixed-point computations and its exten- sion to periodic orbits are discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Finally, the use of Krylov techniques to compute the leading eigenvalues or singular values of the linearized operator to characterize its modal and non-modal stability properties are discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In what follows, we will assume that a time-stepping simulation code is available to simulate the nonlinear system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' the time-stepping code returning Xk+1 = Φτ(Xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Similarly, we will assume that a linearized version of this code can be used to calculate the matrix-vector product Mτx by time-marching the equations, where the operator Mτ is either the (numerically approximated) exponential propagator for fixed points or the monodromy matrix for periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For non-modal stability analysis, we furthermore assume that an equivalent time-stepping code is available to approximate the matrix-vector product M† τx where M† τ is the exponential propagator or monodromy matrix built using the corresponding ad- joint linear operator L†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The methods advocated here and in [82, 156] can all be easily implemented with very few modifications into existing codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 Krylov subspaces and the Arnoldi factorization In [245], the American mathematician Gilbert W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Stewart listed six of the most important ma- trix decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These include the pivoted LU decomposition, the QR decomposition, the spectral (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' eigenvalue) decomposition, the Schur decomposition, and the singular value decom- position (SVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The introduction of each of these decompositions into numerical linear algebra has revolutionized matrix computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Nevertheless, a seventh approximate factorization should be included in this list, namely, the Arnoldi factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Introduced by Walter Edwin Arnoldi in 1951 [9] it relies on the concept of Krylov subspaces [142], named after the Russian applied mathematician Alexei Krylov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Today, these subspaces are the workhorses of large-scale numerical linear algebra and form the foundations of numerous iterative linear solvers such as the minimal residual method (MINRES) [198] or the generalized minimal residual method (GMRES) [222].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The book Iterative methods for sparse linear systems by Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Saad [223] is probably the most complete reference for such techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Given a matrix A ∈ Rn×n and a starting vector x ∈ Rn, a Krylov subspace of dimensions m can be constructed by repeated applications of A, leading to Km(A, x) = �x, Ax, A2x, · · · , Am−1x� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (31) Introducing the matrix K = �x Ax A2x · · Am−1x� , (32) the above sequence can be recast as the following Krylov factorization AK = KC + eT mr, (33) 17 where C ∈ Rm×m is a companion matrix of the form C = � ������ 0 0 0 · · 0 c1 1 0 0 · · 0 c2 0 1 0 · · 0 c3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 0 0 · · · · 1 cm � ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (34) The coefficients ci are computed based on a least-squares procedure such that the residual r is not in the span of the previous m Krylov vectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' rT K = 0, and ∥r∥2 is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' If ∥r∥ = 0, then the columns of K span an invariant subspace, and the eigenvalues of C are a subset of those of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' If the starting vector x is random and m is large enough, K most likely tends towards the invariant subspace of A associated with the eigenvalues having the largest magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Note however that, as m increases, the last Krylov vectors become increasingly collinear by virtue of the applied power iteration and, consequently, the matrix KT K is increasingly ill-conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This companion-based Krylov factorization is thus of little use in practice due to its numerical instability given finite arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A simple remedy to this numerical instability is to iteratively construct each new Krylov vector such that it is orthonormal to all previously generated vectors instead of simply applying the power iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Starting from a vector x1 (with ∥x1∥2 = 1), the Krylov basis can be iteratively constructed by the following algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Algorithm 1 – Arnoldi factorization Given : the n × n matrix A, and a unit-norm vector v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' w = Av1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' α1 = v∗ 1w ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' r = w − v1α ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' V1 = [v1] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' H1 = [α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For j = 1, 2, · · · , m − 1 (a) Add the residual from the previous iteration into the Krylov basis βj = ∥r∥2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' vj+1 = β−1 j r ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Vj+1 = �Vj vj+1 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' ‹ Hj = ï Hj βje∗ j ò ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (b) Compute the residual associated with this new Krylov basis w = Avj+1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' h = V∗ j+1w ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' r = w − Vj+1h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (c) Update the upper Hessenberg matrix accordingly Hj+1 = î‹ Hj h ó .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' After m steps, this leads to the Krylov factorization AV = VH + eT mr, (35) known as the Arnoldi factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In this factorization, the matrix V ∈ Rn×m is orthonormal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' VT V = I) and H ∈ Rm×m is an upper Hessenberg matrix (almost triangular matrix with 18 zero entries below the first subdiagonal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Once again, the residual vector r is the component of the (m + 1)th Krylov vector not in the span of V (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' rT V = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Knowledge of the n × m matrix V and the m × m matrix H can then be used to approximate the dominant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' with the largest absolute value) eigenvalues and eigenvectors of A or to obtain a reasonable solution to Ax = b, at a reduced computational cost compared to direct inversion using Gaussian elimination or LU techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The Arnoldi factorization is at the heart of the widely used GMRES technique for solving large linear systems presented in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Other Krylov factorizations exist such as the Lanczos factorization for Hermitian matrices (where H reduces to a tridiagonal matrix) or the Krylov-Schur factorization introduced by Stewart [244] (where H is in Schur form), enabling simple restarting strategies for computing the dominant eigenvalues and eigenvectors of A when the available RAM is a limiting factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2 Newton-Krylov method for fixed points and periodic orbits For low-dimensional dynamical systems, fixed point (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' periodic orbit) computations can easily be performed using the standard Newton method already implemented in numerous languages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='fsolve in Python).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These implementations (see Algorithm 2) are often quite generic and rely on direct solvers for the inversion of the Jacobian matrix L (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' monodromy matrix Mτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Due to the sheer size of the linear systems resulting from the discretization of par- tial differential equations, this approach however does not scale favorably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Coupling these generic solvers with an existing time-stepping code may moreover require extra layers of code because of the particular data structure used in the simulation and possibly its parallel computing capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Although libraries such as PETSc [14–16] or Trilinos [259] exist, these may also require extra devel- opment to interface with an existing well-established code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' They moreover add extra dependencies which might complicate the deployment of the resulting applications on a large set of computing platforms with different operating systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' from laptops for development to supercomputing facilities for production runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' With the goal of extending the capabilities of an existing time-stepping code with a minimum number of modifications and dependencies, section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2) presents a time- stepping formulation of the Newton-Krylov algorithm to compute fixed points (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' periodic orbits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A similar algorithm has already been introduced by [138] and [82] and in ChannelFlow [111, 112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' As stated previously, we will assume only that we have the nonlinear time-stepper returning Xk+1 = Φτ(Xk), (36) and its linear counterpart xk+1 = Mτxk, (37) where Mτ is either the exponential propagator or the monodromy matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' We will also assume that a routine to compute the m-step Arnoldi factorization (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1) has been implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Along with the inclusion of a direct eigensolver (such as links to LAPACK), this implementation is the only major development needed to extend the capabilities of an existing time-stepping code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Benefits from this implementation outperform its development costs as it paves the way for a GM- RES implementation leveraging all the utilities of the existing time-stepping code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Once available, this m-step Arnoldi factorization routine can also be readily used to compute the leading eigenval- ues and eigenvectors of the high-dimensional linearized operator with no extra development (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 Fixed point computation In a time-stepper formulation, fixed points are solutions to X = Φτ(X), (38) for arbitrary integration time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Alternatively, they are the roots of F(X) = Φτ(X) − X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (39) In fluid dynamics, numerous approaches have been proposed in the literature to compute these fixed points while circumventing the need to implement a dedicated Newton solver into an existing time-stepping code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For example, one can cite the selective frequency damping method (SFD) proposed by ˚Akervik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' [1] and its variants, or BoostConv [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Although they require relatively minor modifications to an existing simulation code, they suffer from a number of limitations, such as slow convergence, which was explored in [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Because it relies on a temporal low-pass filter, the selective frequency damping procedure is, moreover, unable to compute saddle nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' fixed points having at least one unstable eigendirection associated with a purely real eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Assuming the m-step Arnoldi factorization has already been implemented, it requires only a relatively modest effort to integrate it into a dedicated GMRES solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In doing so, the roots of equation (39) can be easily computed using the Newton-Krylov technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The Jacobian of equation (39) is given by J = exp(τL) − I, (40) where L is the linearized operator around the current estimate X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The matrix-vector product Jx thus requires calling the linearized time-stepper (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' to compute exp(τL)x), with x ∈ Rn being the Newton correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Algorithm 2 – Newton-Krylov solver Given : the time-stepper Xk+1 = Φτ(Xk), its linearized counterpart, and an initial guess X0 for the fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For j = 1, 2, · · · , m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Compute the residual of the nonlinear equation r = Φτ(Xj) − Xj 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Check for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' If ∥r∥ ≤ ε, return Xj as the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' If not converged, compute the Newton correction by solving Jx = −r 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Update the solution as Xj+1 = Xj + x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The linear system in step 3 of this iteration is typically solved using a GMRES solver (or other Krylov-based solvers such as BiCGSTAB [268] or IDR [242]) making use of the previously imple- mented Arnoldi factorization (the matrix to be factorized being J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The GMRES procedure is presented in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' It should be emphasized that the linear equation in step 3 does not need to be solved with high precision at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' It is indeed sufficient to ensure that the Newton correction x reduces the norm of the residual r = Φτ(Xj+1)−Xj by one or two orders of magnitude by setting the tolerance in the iterative linear solver to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='01∥r∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Although this may increase the 20 number of Newton steps before convergence is reached, each iteration is computationally cheaper and faster as fewer Krylov vectors need to be generated by the iterative linear solver, thus reducing the overall time to solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Hereafter, such a strategy is referred to as Newton-Krylov with dynamic tolerances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 21 Algorithm 3 – Solving Ax = b with GMRES Consider the linear system Ax = b with A ∈ Rn×n, and both x and b ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' We will assume furthermore that n ≫ 1 so that solving this system with standard direct solvers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' LU factorization) is intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' One of the most famous iterative methods to solve such large-scale systems is the generalized minimal residual method (GMRES), introduced by Yousef Saad and Martin H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Schultz [222] in 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Based on the Arnoldi factorization, it iteratively approximates the solution to Ax = b by the vector xk in a k-dimensional Krylov subspace with minimal residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This task can be formulated as the following optimization problem minimize xk∈K ∥b − Axk∥, where K(r0, A) is a k-dimensional Krylov subspace iteratively constructed using the residual r0 = b − Ax0, with x0 our initial guess for the solution (often taken as the zero vector), and ∥ · ∥ denoting the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Consider now the k-step Arnoldi factorization AVk = Vk+1‹ Hk, with Vk ∈ Rn×k an orthonormal matrix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' VT k Vk = Ik), and ‹ Hk the resulting (k + 1) × k upper Hessenberg matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The unknown vector xk can be expressed as xk = x0 + Vky, with y ∈ Rk an unknown low-dimensional vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The columns of Vk+1 being orthonormal, introducing this expression into the GMRES objective function yields ∥b − Axk∥ = ∥b − A (x0 + Vky) ∥ = ∥r0 − AVky∥ = ∥βv1 − Vk+1‹ Hky∥ = ∥Vk+1 Ä βe1 − ‹ Hky ä ∥ = ∥βe1 − ‹ Hky∥, with e1 the first vector in the standard basis of Rk+1, and β = ∥r0∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The low-dimensional vector y is thus a solution to the following least-squares problem minimize y ∥βe1 − ‹ Hky∥, whose solution is given by y = β‹ H† ke1, where ‹ H† k = Ä‹ HT k ‹ Hk ä−1 ‹ HT k is the Moore-Penrose pseudoinverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' If the residual is too large, a new Krylov vector is generated following the Arnoldi procedure and the iteration continues until ∥βe1 − ‹ Hky∥ is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Numerous variants of GMRES exist, most notably when A is ill-conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For more details, inter- ested readers are referred to the excellent book Iterative methods for sparse linear systems by Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Saad [223].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Let us now consider a critical point in the above formulation of the problem, namely its con- vergence and computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For a reasonable initial guess, the number of Newton iterations is expected to scale as O(k) where k is the number of eigenvalues of Mτ in the vicinity of the 22 unit circle (see [138] for a discussion about the convergence properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In typical production runs, only a handful of eigenvalues may be unstable or close to being unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In our numerical experiments, the Newton solver usually converges in more or less ten iterations, irrespective of the discretization of the underlying partial differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' From a computational point of view, the cost of each Newton iteration is, however, dominated by the call to the GMRES solver where each new Krylov vector is obtained from a linearized simulation over the integration time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This parameter τ plays a crucial role in the number of Krylov vectors that need to be generated to achieve convergence as it directly impacts the eigenvalue distribution of the exponential propagator Mτ = exp(τL) and thus of the Jacobian J = Mτ − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' As τ increases, the gap between the leading eigenvalues and the others increases, and GMRES requires fewer iterations to converge (and thus fewer Krylov vectors must be stored in memory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The wall-clock time of each GMRES iteration however increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Still, a major advantage of the time-stepper formulation is that it does not require preconditioning strategies to perform well (although it can benefit from them, as shown in [82, 263] and the discussion in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A short parametric study was performed to evaluate the performance and sensitivity of the method with respect to the size m of the Krylov subspace and the integration time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' At least three runs4 are performed with m ∈ (50, 75, 100, 125, 150, 175, 200) and τ = (T/12, T/10, T/8, T/6, T/4, T/2, T), where T is the characteristic timescale of the leading eigen- value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This is illustrated in figure 5 for two different flow cases: the two-dimensional cylinder flow at Re = 80 and the two-dimensional open cavity flow at Re = 4700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For the computation of the leading eigenvalues, we can see that similar times to solution are obtained for 4 < mτ T < 20, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' the product of the dimension of the Krylov subspace and the sampling time τ is sufficient to cover between 4 and 20 periods of the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For the cases of non-oscillatory instabilities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' from a pitchfork bifurcation, we suggest a sampling period of τ = 1 non-dimensional time unit as a starting point if no estimate of the doubling time of the instability is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For the cylinder flow, mτ = 67 using dynamic tolerances resulted in a minimum total computation time of about a minute, while mτ = 1128 led to a computation time of over 16 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For the open cavity 2D flow, the minimum total computation time of 30 seconds was achieved with mτ = 10, and the maximum time over 6 minutes was calculated with mτ = 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2 Periodic orbit computation Periodic orbits are solutions to X = Φτ(X) for τ = τ ∗, (41) where τ ∗ is the period of the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' They are the roots of F(X, τ) = Φτ(X) − X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (42) The above system of equations is underdetermined: it has only n equations for n + 1 unknowns (the last unknown being the period of the orbit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' To close the system, an extra phase condition must be considered to select a particular point on the orbit (see Algorithm 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Various possibilities have been suggested in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For example, in AUTO [84, 85] an integral constraint is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 4The tests were computed in a single node of the Jean Zay HPE SGI 8600 supercomputer with two Intel Cascade Lake 6248 processors (20 cores at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 GHz) with the Intel Compiler version 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='4 More information in http: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='idris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='fr/eng/jean-zay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 23 Figure 5: Time to solution (in CPU minutes) versus different pairs of Krylov basis size m and integration time τ for fixed point computation: (a) base flow of 2D flow past a circular cylinder at Re = 80, where the initial condition is assumed to be the base flow at Re = 40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (b) base flow of the 2D open-cavity at Re = 4700 starting from the base flow at Re = 4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Black markers represent time to solution computed with fixed solver tolerances set to 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Red markers represent cases where the tolerance was tightened after each Newton step, from 10−5 at the first step to the target value 10−10 at the final step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The parametric study is carried out using an automated Python script (found in validations/newton_loop/autorun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='py) that loops over previously defined ranges of m and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Here, a simpler condition is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Given an initial condition X0 = X(0), the phase condition is chosen as follows F(X0) · (X − X0) = 0, (43) where F(X0) is the time-derivative of the system evaluated at X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A solution to this bordered system ® Φτ(X) − X = 0, F(X0) · (X − X0) = 0, (44) can be obtained using a Newton-Krylov solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The Jacobian of this system is J = ïMτ − I F (Φτ(X0)) FT (X0) 0 ò .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (45) As before, the corresponding matrix-vector product requires a single call to the linearized time- stepper (which includes many smaller time steps) to evaluate Mτx (see the upper left block of the Jacobian matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Evaluation of the other terms should be readily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 24 (b) (a) 101 minutes] [CPU 100 t 101 102 102 101 mT/T mT/TAlgorithm 4 – Newton-Krylov solver for periodic orbits Given : the time-stepper Xk+1 = Φτ(Xk), its linearized counterpart, and an initial guess X0 for the fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For j = 1, 2, · · · , m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Compute the residual of the nonlinear equation r = Φτ(Xj) − Xj 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Check for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' If ∥r∥ ≤ ε, return Xj as the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' If not converged, compute the Newton correction by solving ïMτ − I F (Φτ(X0)) FT (X0) 0 ò = ï x ∆τ ò .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Update the solution as Xj+1 = Xj + x, τj+1 = τj + ∆τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' When evaluating the matrix-vector product Mτx, both the original nonlinear system and the linearized one need to be marched forward in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' While this increased computational cost is limited for small-scale systems, it may become quite significant for large-scale systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A simple strategy to alleviate this is to precompute the tentative orbit Xk(t) for t ∈ [0, τk] at the beginning of each Newton iteration and store all time steps in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Then, only the linearized system needs to be marched forward in time with its coefficients updated at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' If one is memory- bounded, only a limited number of time steps of the nonlinear system can be stored in memory and the intermediate steps can be reconstructed using for instance cubic spline interpolation or temporal Fourier interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The Newton-Krylov algorithm presented herein corresponds to the standard shooting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' It is by far the simplest method to implement for the computation of periodic orbits in large-scale systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Other techniques exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For instance, multiple shooting [225] leverages the concept of a Poincar´e section while temporal collocation transforms the orbit computation into a boundary value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Recently, Shaabani-Ardali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' [237] have also adapted ideas from feedback control with time delays to stabilize unstable periodic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In figure 6, we show the evolution of the residual with respect to the computational time for stabilization of a periodic base flow with an imposed frequency using the time-delayed technique and the proposed Newton GMRES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' We can observe a striking difference in the residual deflation when comparing the two techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='3 Large-scale eigensolvers Having computed a fixed point or a periodic orbit, one is often interested in its stability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These can be its asymptotic stability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' modal stability characterized by the eigenvalues of the Jacobian matrix L) or short-time stability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' non-modal stability characterized by the singular values of the exponential propagator exp(τL)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' As discussed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1, given a fixed point X∗, the linearized dynamics are governed by dx dt = Lx, (46) where L is the evolution operator linearized about X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For a periodic orbit, these dynamics are 25 Figure 6: Residual deflation as a function of the total computation time (both computed in the same hardware) for time-delayed feedback (open gray circles) and Newton GMRES (filled black diamonds) for the harmonically forced jet presented in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' To obtain a continuous signal over time for the Newton method, we sum the number of calls to the linearized solver between each Newton iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' We observe decays proportional to t−2 for the time-delayed feedback and t−14 for the Newton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' governed by dx dt = L(t)x, (47) with L(t+τ) = L(t) and τ the period of the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In a time-stepper formulation, these continuous- time linear systems are replaced by the following discrete-time one xk+1 = Mτxk, (48) with Mτ being the exponential propagator or monodromy matrix, depending on the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The matrix-vector product Mτxk amounts to integrating forward in time the linearized equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' While for a fixed point, Mτ and L have the same set of eigenvectors, their eigenvalues are related by λi = log(µi) τ , (49) where τ is the sampling period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For a periodic orbit, the eigenvalues µi of Mτ are directly the Floquet multipliers needed to characterize the stability of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In both cases, the leading eigenpairs of Mτ can easily be computed using the Arnoldi factor- ization described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 and algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Consider the factorization MτV = VH + βeT mr, (50) with V ∈ Rn×m an orthonormal Krylov basis, H ∈ Rm×m an upper Hessenberg matrix and r the unit-norm residual after m steps of the Arnoldi iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Introducing the ith eigenpair (ˆµi, yi) of the m × m upper Hessenberg matrix into the Arnoldi factorization leads to ∥MτVyi − Vyiˆµi∥2 = |βeT myi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (51) Hence, if the left-hand side is small enough, the pair (ˆµi, Vyi) provides a good approximation of the ith eigenpair of the operator Mτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' If one is interested in the short-time stability properties instead, 26 10-2 10-4 10-6 E 10-8 10-10 10-12 10-2 10-1 100 101 102 CPU hoursAlgorithm 5 – Solving Ax = λx with the Arnoldi factorization Consider the eigenvalue problem Ax = λx, with A ∈ Rn×n, x ∈ Cn and λ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The k-step Arnoldi factorization reads AVk = VkHk + βvk+1eT k , with Vk ∈ Rn×k an orthonormal matrix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' VT k Vk = Ik), Hk the k × k upper Hessenberg matrix, and ek the kth vector in the standard basis of Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Given the ith eigenpair (λi, yi) of the Hessenberg matrix, the ith eigenpair of the original matrix A can be approximated as λi, and xi = Vkyi, with the low-dimensional eigenvector yi being normalized such that ∥yi∥2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The residual associated to this approximate eigenpair (also known as a Ritz eigenpair) is given by ∥Axi − λixi∥ = ∥AVkyi − λiVkyi∥ = ∥Vk (Hk − λiI) yi + vk+1βeT k yi∥ = |β||eT k yi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' If this residual is small enough for a sufficiently large number of Ritz eigenpairs, then the computation stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Otherwise, a new Krylov vector is added to the basis V, and the Arnoldi factorization continues until the desired number of Ritz eigenpairs have converged below the user-defined tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' the leading singular modes and associated gain can be computed using the same algorithm where Mτ is replaced by M† τMτ = exp�τL†� exp (τL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' As for the Newton-Krylov solver presented in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2, the computational time is dominated by that of the call to the linearized solver needed to compute the matrix-vector product Mτxk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For fixed points, the choice of τ is also important as it plays a key role in the spectral gap between the eigenvalues of interest and the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This is discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The initial vector v1 used to generate the Krylov subspace is also important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Assuming random white noise distributed in the perturbation fields, the eigenpairs effectively start to converge only after a sufficiently large number of Krylov vectors have been generated such the transients are washed out of the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The number of Krylov vectors that must be generated before this happens is, of course, dependent on the sampling period τ and the size of the computational domain under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' From our experiences, assuming only one eigenvalue is unstable a good trade-off in terms of time- to-solution and computational cost is obtained when one chooses the size m of the Krylov subspace and the sampling period τ such that mτ T = O(10), where T is an a priori estimate of the typical timescale of the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' If multiple eigenvalues are unstable, T can be selected as the slowest time-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' We point out that such estimates should be regarded as indicative only, since effective computational performance depends on many other parameters not considered here (hardware, operating system, compiler, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Figure 7 shows a parametric study performed to evaluate the performance and sensitivity of the method with respect 27 to the size m of the Krylov subspace and the integration time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' At least three runs were made with m ∈ (50, 75, 100, 125, 150, 175, 200) and τ = (T/12, T/10, T/8, T/6, T/4, T/2, T), where T is the characteristic timescale of the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Two cases are considered: the two-dimensional cylinder flow at Re = 80 and the two-dimensional open cavity flow at Re = 4700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Figure 7: Time to solution (in CPU minutes) versus different pairs of Krylov base size m and integration time τ for the computation of eigenvalues: (a) 2D flow past a circular cylinder at Re = 80 (T = 1/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='125);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (b) 2D open-cavity at Re = 4700 (T = 1/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='676).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Red markers represent the time to solution for the convergence of 4 eigenmodes and black markers are for 40 eigenmodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For each value of mτ three runs with different initial conditions are computed to account for fluctuations of the computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Eigenvalues with residual lower than 10−6 are considered converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The parametric study is carried out using an automatic python script (located in validations/ eigen_loop/autorun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='py) looping over previously defined ranges of m and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 4 Examples This section illustrates different applications of the use of Krylov methods to study large-scale dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' All examples are taken from fluid dynamics applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Numerical simula- tions rely on the spectral element solver Nek5000 [80, 100] and the dedicated open-source toolbox nekStab nekstab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='io (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' It should be noted that, although we have focused our attention on a particular CFD solver, the methods presented earlier are quite general and can be relatively easily implemented in other partial differential equation solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' We give a brief physical description of each case, as well as details of the base flow and stability calculation, and a compar- ison with a reference work from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Finally, a brief bifurcation analysis is presented for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' All the files needed to run these examples can be found in the nekStab/examples folder, available in the repository github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='com/nekStab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 The flow in a two-dimensional annular thermosyphon Under the influence of unstable thermal stratification, and for a range of control parameters, the two-dimensional flow in an annular thermosyphon is perhaps one of the simplest and cheapest computational test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The geometry considered is the same as in [157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' It consists of two concentric circular enclosures, the inner radius being R1 and the outer radius R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The ratio of the 28 (b) (a) 102 101 [CPU t 100 100 102 102 101 101 mT/T mT/Touter to inner radius is set to R2 R1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A constant temperature T0 is set at the upper walls, while the lower ones are set at a temperature T1 = T0 + ∆T, with ∆T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Hereafter, we work with the non-dimensional temperature ϑ(x, y) defined as ϑ(x, y) = T(x, y) − T0 ∆T , where x and y are the horizontal and vertical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The origin of our reference frame is chosen to be the center of the thermosyphon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Using this non-dimensionalization, the temperature at the lower walls is thus ϑw(y < 0) = 1, while the temperature at the upper ones is ϑw(y > 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Gravity acts in the vertical direction, along −ey, and is characterized by the gravitational acceleration g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Assuming the working fluid is Newtonian, it is characterized by its density ρ, its dynamic viscosity µ, its thermal expansion coefficient β, and its thermal diffusivity α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Using ∆T as the temperature scale, and R2 −R1 as the length scale, we can define two non-dimensional parameters, namely the Rayleigh number Ra = ρgβ∆T (R2 − R1)3 µα , and the Prandtl number Pr = ν α, where ν = µ/ρ is the kinematic viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For this example, the Prandtl number is set to Pr = 5, and the Rayleigh number is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For simplicity, the flow is assumed two-dimensional and incompressible, so that the effect of density variations due to temperature can be modeled using the Boussinesq approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Under these assumptions, the dynamics of the flow are governed by the following Navier-Stokes equations ∂u ∂t + ∇ · (u ⊗ u) = −∇p + Pr∇2u + RaPrϑey, ∂ϑ ∂t + (u · ∇) ϑ = ∇2ϑ, ∇ · u = 0, where u(x, t) is the velocity field, p(x, t) the pressure field, and ϑ(x, t) the temperature field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The computational domain is discretized using 32 spectral elements uniformly distributed in the azimuthal direction, and 8 elements uniformly distributed in the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Within each element, Lagrange interpolation of order N = 7 based on the Gauss-Lobatto-Legendre quadra- ture points is used in each direction, resulting in 16 384 grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Temporal integration is performed using a third-order accurate scheme, and the time step has been chosen to satisfy the Courant–Friedrichs–Lewy (CFL) condition with Courant number Co < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 for all the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Despite the lack of turbulent dynamics due to the absence of the vortex stretching mechanism, the flow configuration can exhibit Lorenz-like chaotic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This low-cost case follows the same structure as the bifurcation diagram of the Lorenz system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 Pitchfork bifurcation The first bifurcation of the flow occurs at the critical Rayleigh number Rac,1 ≃ 494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' It is associated with symmetry breaking in the temperature distribution, which leads to the emergence of a station- ary convection cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The calculations presented were performed with τ = 1 (diffusive time unit) and 29 Figure 8: Eigenvalues for the destabilization of the fixed point for the thermal convection on an annular thermosyphon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Figure 9: The flow in a two-dimensional annular thermosyphon: (a) temperature field (ϑ) of the bilateral symmetric fixed point and (b) real part of the temperature field (Re(ϑ)) of the unstable steady mode at Ra ≃ 499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' a Krylov subspace of dimension m = 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The choice of the sampling period for stationary modes is not straightforward, as a small value of τ leads to a poorly conditioned basis and the generation of spurious modes, while a large sampling period leads to an excessive computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The base flow, corresponding to pure conduction, is shown in figure 9(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The eigenvalue spectrum in fig- ure 8 shows a purely real eigenvalue stepping into the upper half-complex plane for Ra ≥ 494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The associated eigenvector is depicted in figure 9(b) and leads to symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The correspond- ing bifurcation is thus a pitchfork bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Above Rac,1, the conduction-dominated symmetric base flow is no longer stable and is replaced by a convection cell as shown in figure 11(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' From symmetry considerations, this convection cell is equally likely to be associated with clockwise or counterclockwise flow (as shown in figure 11(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2 Hopf bifurcation The new base flow is stable over a wide range of Rayleigh numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' It eventually becomes un- stable at Rac,2 ≃ 16 081 through a Hopf bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This is indicated by a pair of complex conjugate eigenvalues of the associated linearized Navier-Stokes operator moving toward the upper half-complex plane in figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The spatial structure of the leading mode is shown in figure 11(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2 - Ra = 510 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='0 6 Ra = 499 A Ra = 490 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2 - Ra = 480 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='0 0 (a) (b) 2 90 2 2 0 2 2 0 2Figure 10: Eigenvalues of the flow of the steady convection cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The dashed line represents the frequency f = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='67 from a DNS at Ra = 16100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Figure 11: The flow in a two-dimensional annular thermosyphon: (a) temperature field (ϑ) of an unstable steady convection cell and (b) real part of the temperature field (Re(ϑ)) of the unsteady unstable mode at Ra = 16100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The convection cell shown in figure 11(a) starts to oscillate with a characteristic Strouhal number St = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The frequency predicted by the linear analysis and the frequency measured in our DNS match very well near the bifurcation point when the nonlinear distortion of the base flow is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Eigenvalue calculations were performed with τ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='014 (corresponding to a standard recom- mendation for the sampling period of τ = T/8) convective time and using a Krylov subspace of dimension m = 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Base flows were calculated using the Newton-Krylov method under the same conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For comparison metrics, the calculation of the unstable base flow at Ra = 16 100 start- ing from the base flow at Ra = 16 000 took 147 seconds with Newton-Krylov compared to 1071 seconds with selective frequency damping (SFD) [1] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' one seventh of the time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For SFD, we considered a parameterization proposed by [61] that leads to a more robust selection of the cutoff and gain for low-pass filtering compared to the original guidelines [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='05 Ra = 16100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='00 b Ra = 16082 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='05 Ra = 16050 Ra = 16000 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='10 口 10 0 5 10 5 f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='0 0 (a) (b) 2 9o 2 2 0 2 2 0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2 The harmonically forced jet Our attention now shifts towards a time-periodic flow harmonically forced via the inflow boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The dynamics of the flow are governed by the incompressible Navier-Stokes equations ∂u ∂t + ∇ · (u ⊗ u) = −∇p + 1 Re∇2u, ∇ · u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (52) The Reynolds number is defined as Re = U0D ν , where U0 is the velocity at the jet centerline, D the jet diameter, and ν the kinematic viscosity of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For simplicity, the jet is assumed to be radially symmetric, and the axisymmetric formulation of the Navier-Stokes is considered with x = (z, r), z representing the streamwise direction and r the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The time-periodic structure is forced via a Dirichlet inflow boundary condition prescribed as u(z = 0, r, t) = 1 2 ß 1 − tanh ï 1 4θ0 (r − 4r−1) ò™ (1 + A cos(ωft)), with A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='05 the force amplitude, θ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='025 the initial thickness of the dimensionless shear layer, and angular frequency ωf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The non-dimensional frequency is the Strouhal number St = ωfD/(2πU0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The computational domain extends from 0 to Lz = 40 in the streamwise direction and 0 to Lr = 5 in the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The domain is discretized with nz × nr = 160 × 30 spectral elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Lagrange interpolants of order N = 5 are considered, resulting in 172 800 grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Based on the recent work of [238], we reproduced a case with St = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='6 and investigated its modal stability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Under these conditions, a forced limit cycle is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For subcritical conditions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Re < Rec), vortices form periodically along the shear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' They are then transported in the streamwise direction before fading out because of viscous diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' No pairing phenomena are identified despite the appearance of harmonics in the velocity signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Above the critical Reynolds number Rec = 1371, the vortices spontaneously start to pair, forming larger vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The vortex pairing is connected to a subharmonic instability created via a period-doubling bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 Period-doubling bifurcation In [238], the time-delayed feedback technique [237] was used to stabilize periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The technique is based on the nonharmonic component filtering approach introduced by [210] using an optimal filter gain derived in [237].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Here, the (unstable) time-periodic base flow is computed using the Newton-Krylov method with dynamic tolerances (introduced in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1), with τ = 1/St and a Krylov subspace dimension of m = 128 until a residual level of 10−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The same parameters are used for the Floquet analysis, which is sufficient to converge 20 eigenpairs to a precision of 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' An unstable base flow without vortex pairing can be seen in figure 12(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Figure 12(b) illustrates the dominant Floquet mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Figure 13 shows the spectrum of Floquet multipliers, with the leading one leaving the unit circle along µ = −1, which is characteristic of a period-doubling bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The critical number Rec,1 ≃ 1371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='18 is in excellent agreement with the reference value Rec,1 ≃ 1371 given in [238].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Despite the strong non-normality of the system operator, the Floquet 32 Figure 12: The harmonically forced jet: (a) vorticity component of the stabilized unstable limit cycle at supercritical Re = 2000 and (b) spatial distribution of the subharmonic Floquet mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Inflow forcing with 5% amplitude and StD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' analysis accurately predicts the leading mode responsible for the vortex pairing mechanism observed in nonlinear simulations [238].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' We verified the predictions with nonlinear simulations at subcritical Re = 1370 and supercritical Re = 2000, as shown in figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' At Re = 1370 (in black), the time series of the velocity probe and its Fourier spectrum are characterized by the forced frequency St = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='6 and its harmonics formed by nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' At Re = 2000 (in red), one can see the response of the flow in the velocity signal with the sharp increase in the subharmonic frequency at St = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='3, showing the growth of the secondary instability through a period-doubling bifurcation and the increase in the period of the limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='3 The flow past a circular cylinder Let us now consider the example of a canonical cylinder flow assumed to be infinite in the spanwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The dynamics of the flow are governed by the Navier-Stokes equations (52), with the Reynolds number defined on the basis of the free-stream velocity and the cylinder’s diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The two-dimensional mesh considered in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 is made of 1464 spectral elements (66 in the flow direction and 30 in the vertical), all of which are comparable to previously reported domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' To limit the computational cost, we consider Lagrange interpolants of order N = 5, which shows good convergence compared to N = 7 which leads to a total of 52 704 grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For the three- dimensional problem considered in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2, this mesh is extruded in the third direction, using 10 elements in the spanwise direction to a length Lz = 2π/βc ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The total number of grid points is then 3 162 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' As for the other examples, we consider a third-order accurate temporal scheme and a time-step that satisfies Co < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 for all simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 Primary instability and sensitivity analysis At low Reynolds numbers, the flow is two-dimensional, steady and symmetric with respect to the cross-stream direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' According to Jackson [131] and more recently Kumar & Mittal [144], the flow is expected to become unstable at Rec,1 = UD/ν ≈ 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='6, leading to the emergence of the well-known von K`arm`an vortex street characterized by Stc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This primary instability is a canonical example of a supercritical Hopf-type bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Numerous wake flows [206] exhibit an instability that leads to the onset of periodic vortex shedding and their characterization as flow oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 33 (a) 0 2 3 5 0 (b) 2 0 3 3 1 0 0 5 10 15 20 25 30 35 40 ZFigure 13: Evolution of Floquet multipliers of the harmonic forced axisymmetric jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The single leading unstable eigenmode is associated with the vortex pairing mechanism and a period-doubling bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The almost superposed black star represents the reference value from [238].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Figure 14: Radial velocity signal at x, r = (5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5) at subcritical Re = 1370 (black) and supercritical Re = 2000 (red): (a) signal evolution and (b) discrete Fourier transform (DFT) spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (c) Phase portrait of the system in coordinates (v, ˙v, ¨v): one can observe the departure of the system trajectory from the stable limit cycle (in black), timidly exploring the phase space before settling into a period-doubled orbit (in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 34 Re = 1300 Re = 1370 Re = 1375 Re = 2000 1 0 1(a) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2 20 40 60 80 0 100 120 140 七 (b) 105 St1 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' St2 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='30 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='0 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2 10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='3 2 1 3 5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 2: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='0 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='0Sensitivity to base flow changes based on the linear Navier-Stokes operator was introduced by Bottaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' [37] in a local framework and was later extended to the global framework by Marquet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' [174].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Base flow sensitivity analysis has been shown to provide valuable information for shape optimization or actuator placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The sensitivity analysis of the complex eigenvalue λ = σ + iω, with the real part σ being the growth rate of the eigenmode and the imaginary part ω its frequency, allows the computation of vector fields that highlight the net effects of generic small-amplitude base flow modifications δU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Both the sensitivity analysis for generic modifications of a base flow (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' U) and for a steady force (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' F) have been implemented in nekStab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Structural changes in the complex eigenvalue δλ due to small-amplitude arbitrary base flow modifications δU can be formally related through the inner product δλ = ⟨∇Uλ|δU⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The gradient ∇Uλ is a complex vector field that defines the sensitivity to base flow modifications, given by ∇Uλ = −(∇u)H · u† + ∇u† · ˆu∗, (53) with the superscript H representing the conjugate transpose, † the adjoint, and ∗ the complex conjugate (for a complete derivation, the reader is referred to Marquet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' [174]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The first term of equation (53) is the sensitivity to transport modifications, while the second term is the sensitivity to production modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Figure 15 depicts the real and imaginary parts of ∇Uλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These represent the sensitivity of the growth rate of the eigenvalue to base flow modifications, and the frequency sensitivity of the eigenvalue to base flow modifications, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Generic base flow modifications in negative sensitivity zones promote stabilization of the eigenvalue (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' reduction of the growth rate or frequency), whereas changes in positive sensitivity zones promote destabilization or frequency increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Analogously, the sensitivity to a steady force can be derived by introducing an inner product in relation to a volume force F in the form δλ = ⟨∇Fλ|δF⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The complex sensitivity function ∇Fλ is given by the knowledge of adjoint base flow fields in the form ∇Fλ = U†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (54) This adjoint base flow is a solution to the linear system of equations −∇U† · Ub + (∇Ub)T · U† − ∇P † − 1 Re∇2U† = ∇Uλ, ∇ · U† = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (55) Assuming the adjoint Jacobian matrix L† has already been projected onto a divergence-free sub- space, this equation can be written as L†U† = ∇Uλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Note that, in a time-stepper framework, we do not have access to L†, but only exp�τL†�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Accounting for the time derivative, the general solution of the linear system above is given by U†(τ) = exp Ä τL†ä U†(0) + � τ 0 exp Ä (τ − t)L†ä ∇Uλ dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 35 Figure 15: Streamwise component of the sensitivity to base flow modifications ∇Uλ of the leading eigenvalue λ at Reynolds number Re = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Spatial distribution of (a) the growth rate sensitivity ∇Uσ and (b) the frequency sensitivity ∇Uω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In a time-stepper framework, equation (55) is thus replaced with Ä I − exp Ä τL†ää U† = � τ 0 exp Ä (τ − t)L†ä ∇Uλ dt, (56) which is solved using the GMRES solver discussed earlier in the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Note that the right- hand side is computed by running an adjoint simulation with the steady force ∇Uλ for τ time units with a zero initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Figure 16 depicts the real and imaginary parts of ∇Fλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These are the sensitivity of the growth rate of the eigenvalue to changes due to a steady force, and the associated frequency sensitivity of the eigenvalue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These maps are consistent with those obtained in [174] and experimentally in [246].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Although not reproduced here, the theoretical framework was extended by Giannetti, Camarri, and Citro [109] to include sensitivity analysis with respect to generic modifications and a force acting on periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2 Pitchfork bifurcation We now focus on the Floquet analysis of three-dimensional modes evolving in a two-dimensional time-periodic base flow formed in the wake of a circular cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This problem has been studied in detail by Barkley & Henderson [19] on the basis of a Fourier expansion in the spanwise direc- tion, as is the gold standard for demonstrating a (secondary) pitchfork bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In addition, this symmetry-breaking bifurcation [189] is characterized by a single real eigenvalue that becomes positive, corresponding to a single Floquet multiplier that leaves the unit cycle by µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' At this point, the flow experiences a steady bifurcation (synchronous with the underlying periodic 36 Vuo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 9 1 Vuf (b) 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='18 0- 9 1 2 2 0 4 6Figure 16: Normalized modulus of the sensitivity to a steady force ∇Fλ of the leading eigenvalue λ at Reynolds number Re = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Spatial distribution of (a) the growth rate sensitivity ∇Fσ and (b) the frequency sensitivity ∇Fω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 37 VFO 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 1 9 0 1 VFf (b) 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 0- 9 1 2 2 0 4 6orbit), thus not altering the temporal structure of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The frequency of the underlying limit cycle remains the same, but the spanwise invariance of the flow is broken, resulting in a three- dimensionalization of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Barkley & Henderson [19] report a secondary instability to what is called mode A occurring at Rec,2 ≃ 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5±1 with a critical wavenumber βc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='585 in the spanwise direction, corresponding to a length of nearly four diameters in the spanwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Using a domain (Li, Lh, Lo) = −15D, ±22D, 25D, they report a limit cycle oscillating with St = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1954 at Re = 190, and a leading synchronous Floquet mode with µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Later Giannetti, Camarri, and Luchini [110], employing a finite element code and a domain (Li, Lh, Lo) = −16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5D, ±11D, 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5D, calculated Rec,2 ≃ 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='77 and some reference values at Re = 190 for the limit cycle frequency St = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1971 and µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='002 for mode A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Recently, Giannetti, Camarri, and Citro [109], using a spectral element code and a domain (Li, Lh, Lo) = −15D, ±15D, 35D, obtained Rec,2 ≃ 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='71 and reported a limit cycle with St = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1962 at Re = 190, as well as µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='009 for mode A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Barkley [21] mentions that accurate estimates of Floquet stability analyses can be obtained in smaller domains (Li, Lh, Lo) = −8D, ±8D, 25D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Since nekStab relies on the spectral element solver Nek5000, which does not use a Fourier decomposition in the spanwise direction, a finite-span domain is specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A limit cycle can be computed directly in the 3D domain, or due to the geometry of the flow computational time can be saved simply by setting the spanwise velocity component to zero or by extruding a 2D solution into a 3D mesh (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' using the pymech [54] package).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In our case, using the Newton-Krylov method for stabilizing UPOs, we first perform a DNS at subcritical Re = 187 and thus transients from the initial condition are convected out of the computational domain, obtaining a 2D subcritical periodic orbit in the 3D domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The orbit period is measured from the DNS and used as an initial guess for the Newton GMRES algorithm, together with a snapshot of the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In this way, we can gradually stabilize the UPOs at supercritical Reynolds numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A Krylov subspace of dimension m = 64 is used, both for base flow and Floquet stability calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Due to the almost degenerate nature of the eigenvalues, a Krylov-Schur iteration is used for the convergence of at least 4 direct modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The evolution of Floquet multipliers moduli is shown in figure 17, as well as the best linear fit for their evolution along the Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The fully 3D leading unstable Floquet mode at Re = 190 is shown in figure 18 and shows excellent qualitative agreement with the unstable mode presented in Barkley & Henderson [19] and also in Blackburn and Lopez [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Our estimate for the critical Reynolds number Rec,2 = 189 using a domain with βc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='585 agrees very well with the range of values reported by Barkley & Henderson [19] as well as Giannetti, Camarri and Luchini [110], despite our necessity to fix a spanwise wavenumber that could not precisely match the critical one reported, as well as a different mesh strategy, domain size, and polynomial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Specifically, our µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='012 at Re = 190 differs by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='15% with µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='034 by Barkley [21] and 1% with µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='002 by Giannetti, Camarri, and Luchini [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='4 The flow past side-by-side circular cylinders The flow past two side-by-side cylinders is mainly governed by a Reynolds number based on the cylinder diameter and the free-stream velocity (similar to a single-cylinder wake), but with the addition of a separating distance (gap) between the surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A mesh with 5092 elements and polynomial order N = 7 is considered, spanning from -50 to 75 in the streamwise direction and 50 to 50 in the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Taking into account a gap g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='7, [59] reports a primary Hopf bifurcation at Rec,1 ≃ 55 with a synchronized vortex shedding wake oscillating at Stc,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' At Rec,2 ≃ 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='7, a secondary mode known as the “flip-flop” mode with Stc,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='02 becomes unstable due to a supercritical Neimark-Sacker bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The flow is bistable: an asymmetric dual wake arises, in which much slower switching can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 38 Figure 17: Modulus of the dominant Floquet multiplier as a function of the Reynolds number for the flow past a circular cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 Neimark-Sacker bifurcation The Neimark-Sacker or secondary Hopf is characterized by the emergence of a new incommensurate frequency in the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This bifurcation differs from a pitchfork when the instability created oscil- lates with the same frequency of the periodic orbit and from period-doubling when the instability created is subharmonic with respect to the periodic orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Figure 19 shows the spectrum of Floquet multipliers for different Reynolds numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The critical Reynolds number is Rec,2 ≃ 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='17, which is in excellent agreement with the reference value Rec,2 ≃ 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='6 reported in [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The mode leaves the unit disk at an angle of 71 degrees (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' φ = tan−1(Im(µ)/ Re(µ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Figure 20 shows a snapshot of the vorticity distribution of the UPO under supercritical conditions Re = 67 and the vorticity distribution of the unstable Floquet mode associated with the flip-flop mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Figure 21(a,b) show the time trace and the Fourier spectrum of the velocity recorded by a probe in the wake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The main discrete Fourier transform (DFT) spectrum peak is located at the fundamental frequency of the UPO, with another (second) peak occurring at the new incommensurable frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Fig- ure 21(c) shows the phase-space representation of dynamics with the formation of a torus object characteristic of quasiperiodic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 Backward-facing step The flow past a backward-facing step is chosen as an example of a transient growth analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' We reproduce the analysis presented in [35] at Re = 500 and take the step height as the charac- teristic length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' As before, the dynamics are governed by the incompressible Navier-Stokes equations (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' We considered a mesh made up of 1670 elements, each locally discretized with a polynomial order N = 5 in both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The domain spans from -10 to 50 in the streamwise direction, and from -1 to 1 in the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' At Re = 500, the stationary base flow is linearly stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Yet, small perturbations can experience large transient growth due to the strong non-normality of the linearized Navier-Stokes operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The maximum gain envelope is calculated and compared with the reference in figure 22, showing excellent agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The peak is located at τ = 58, which corresponds to the maximum possible transient growth associated with the optimal initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Figure 23 shows the fixed point computed by the Newton-Krylov method, the spatial distribution of the leading eigenvector of the direct-adjoint eigenproblem corresponding to the optimal perturbation with maximum energy gain, and the optimal response at target time 39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='04 lμl(Re) ～ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='008965Re - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='694566 Barkley & Henderson (1996) Giannetti, Camarri and Citro (2019) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='02 Giannetti, Camarri and Luchini (2010) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='98 187 188 189 190 ReFigure 18: The flow past a circular cylinder: semitransparent vorticity magnitude contours (ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='35) of the limit cycle at supercritical Re = 190 superimposed with streamwise vorticity contours (ωx = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='18) of the real part of the unstable steady mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' τ = 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Excellent agreement with reference [35] is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 5 Discussion In this section, we discuss various practical and theoretical aspects of using a time-stepper formula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This includes the interpretation of time-stepping as an effective preconditioner in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 and some observations on the convergence of leading eigenvalues for the linearized Navier-Stokes operator in open shear flows in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 Time-stepper and preconditioning Solving a linear system forms the most computationally intensive part of Newton solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Using a standard approach, one needs to solve linear systems of the form Lx = b, where L ∈ Rn×n is the Jacobian matrix of the Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' For the sake of this discussion, we will assume furthermore that n is sufficiently large so that direct solvers cannot be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' No matter the discretization technique employed, this matrix tends to be ill-conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Consequently, directly solving the above system of equations with an iterative solver may lead to rel- atively poor computational performances as the convergence rate of such solvers is directly impacted by the conditioning of the matrix [223].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' It should be emphasized furthermore that explicitly com- puting this matrix-vector product might not moreover be easily accessible given a general-purpose CFD solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In order to overcome the first issue, practitioners typically use preconditioning, either left preconditioning leading to a new system of the form PLx = Pb, or right preconditioning, leading to LPy = b, and Py = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 40 Figure 19: Evolution of Floquet multipliers for the flow past two side-by-side cylinders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A pair of modes associated with the flip-flop instability leaves the unit cycle increasing its moduli (both growth rate and frequency) as a function of Re in a Neimark-Sacker bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The black star represents the reference value from [59] at Re = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Figure 20: The flow through side-by-side circular cylinders: (a) snapshot of the limit cycle and (b) Floquet “flip-flop” mode at Re = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 41 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 X LO 区 X C Re = 60 Re = 62 Re = 63 Re = 67 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 R(μ)a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='0 0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 0 10 15 20 5Figure 21: Vertical velocity signal at (x, y) = (5, 0) in the flow past side-by-side cylinders with g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='7 for the subcritical Re = 60 (black) and supercritical Re = 62 (dark red) regimes: (a) signal evolution and (b) DFT spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (c) trajectory of the system in phase space using the coordinates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' v, ˙v) of both the limit cycle (black) and the torus (dark red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Figure 22: Envelope of the optimal gain for the 2D backward-facing step problem computed for Re = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The parametric study is carried out using an automatic Python script (located in examples/backward_facing_step/autocomp_tg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='py) looping over a predefined range of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 42 (a) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='10 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 0 100 200 300 400 500 600 700 800 t 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='00 (b) 105 St1 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1113 St1 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='05 St2 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='0206 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='10 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='02 +S×104 Barkley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (2008) 6 2 20 40 60 80 100 0 tFigure 23: The backward facing step problem: (a) fixed point at Re = 500, (b) optimal disturbance, and (c) optimal response at the maximum amplification time horizon τ = 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In both cases, the matrix P ∈ Rn×n (known as the preconditioner) should be a reasonably good, and more importantly cheap, approximation of the inverse of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A judicious choice of P can lead to a substantial reduction of the number of steps needed for the iterative solver to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This choice however strongly depends not only on the spectral content of L but also on its structure directly inherited from the discretization scheme employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The most standard preconditioners include Jacobi preconditioning, block LU, additive Schwarz methods, or algebraic and geometric multigrids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In fluid dynamics, preconditioners designed specifically for fixed point computation using a Newton solver include the Stokes and Laplace preconditioning by Tuckerman and col- leagues [20, 45, 264, 265], or their extensions by Gelfgat [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Given a classical time-stepping solver for the incompressible Navier-Stokes equations, these are relatively easy to implement (see more discussions in [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Yet, to the best of our knowledge, no systematic methods are however available to choose a priori the best preconditioner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In contrast, the linear system involved in the time-stepper formulation of the Newton-Krylov method reads (exp(τL) − I) x = b, where the sampling period τ plays a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' At first, not much seems to be gained from this formulation as, for a sampling period τ comparable to the discretization time step ∆t, we have exp(τL) − I ≃ (I + τL) − I ∝ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Hence, for a small sampling period, the condition number of exp(τL) − I is directly proportional to that of the Jacobian matrix L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' However, in practice, τ is of the order of a few hundred or a few thousand time-steps ∆t (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' τ ≫ ∆t) such that the first-order Taylor approximation of the matrix exponential does not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' An upper bound on the number of iterations needed to drive the residual ∥rk∥2/∥r0∥2 below a given tolerance ε can be derived analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Assume for the moment that all the eigenvalues of L lie in the stable complex half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Furthermore, we will assume that the leading eigenvalue λ1 satisfies Re(λ1) ≤ −δ, with δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The spectrum of the matrix J = exp(τL) − I then satisfies spec(J) ∈ D = {z : |z + 1| ≤ exp(−τδ)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 43 (a) 9 0 (b) 0 0 (c) 90 5 15 5 0 10 20 25 30 35Adapting the derivation in [223] of the general purpose GMRES method to our particular case, it can be shown that the number k of iterations needed to drive the normalized residual below a given tolerance ε satisfies ε ≤ κ(V) exp(−kτδ), where V denotes the matrix of eigenvectors of J (which are identical to the eigenvectors of L), and κ(V) = ∥V∥2∥V−1∥2 its condition number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' From this expression, we can then write k ≤ 1 τδ log Åκ(V) ε ã .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' It is well known that such upper bounds tend to be overly pessimistic, and typically are worst-case scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' It nonetheless highlights the fact that increasing the sampling period τ is likely to reduce the number of GMRES iterations needed for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This is consistent with the fact that, if L has only stable eigenvalues, then lim τ→∞ exp(τL) − I = −I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Moreover, when exp(τL) has p eigenvalues outside of the unit disk, it can be shown that GMRES only needs O(p) extra iterations to converge, and this upper bound remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Note however that, as we increase τ, the computational cost associated to generating each Krylov vector also increases, and a trade-off needs to be found between the computational cost of generating a Krylov vector and the memory footprint of storing a large Krylov basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' It should be noted furthermore that, while exp(τL) − I is well conditioned, this is obtained at the cost of taking many small timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In contrast, the operator in [20, 264, 265] is not as well conditioned, but consists of a single timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This trade-off between fewer GMRES iterations for a well-conditioned but costly operator vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' more GMRES iterations for a less well-conditioned but inexpensive operator is discussed in detail in [263].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' From a practical point of view, using a time-stepper formulation can nonetheless be understood as an effective and easy-to-use preconditioning strategy as already pointed out by Tuckerman and collaborators [20, 264, 265].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2 Observations about the convergence of eigenvalue computations Let us now discuss some practical aspects associated with eigenvalue computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These include some general statements about the convergence of the leading eigenvalues, the choice of the sampling period τ, as well as practical considerations regarding the influence of the streamwise extent of the domain for open flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='1 General statements about the convergence of the Arnoldi iteration Suppose the matrix exp(τL) ∈ Rn×n is diagonalizable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' exp(τL) = VDV−1, with Dii = µi, and consider the orthonormal matrix Q ∈ Rn×k obtained after a k-step Arnoldi factorization started with the unit-norm vector ˆb = b ∥b∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The vector ˜b can be expressed as the linear combination of the eigenvectors such that ˆb = n � i=1 αivi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 44 It can be shown that the approximation of the ith eigenvector vi in the Krylov basis Q satisfies ���� Ä I − QQT ä vi ���� ≤ Ü n � j=1 j̸=i |αj| |αi| ê ε(k) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (57) A detailed derivation of this statement can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In the expression above, ε(k) i is given by ε(k) i = min p∈Pk−1 p(µi)=1 max(|p(µ1)|, · · · , |p(µn)|), where Pk−1 denotes the set of polynomials of degree k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Two key observations can be derived from this upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Fast convergence for isolated eigenvalues Suppose µi is an isolated eigenvalue such that the rest of the spectrum can be enclosed in the disk D = {z : |z − c| < ρ} , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' a disk centered in c and of radius ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In a time-stepping framework, the center c typically is the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Following [223], an upper bound for ε(k) i is given by ε(k) i ≤ κ(V) Å ρ |µi − c| ãk , where κ(V) ≥ 1 is the condition number of the matrix of eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Clearly, the farther away µi is from the rest of the spectrum, the smaller this upper bound, and hence the faster the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This observation is of utmost importance in numerous fluid dynamics applications where branches of eigenvalues are pretty common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These branches are notoriously associated with the non-normality of the linearized Navier-Stokes operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Not only do they impact directly on the condition number κ(V), but they are also non-isolated eigenvalues which are thus much harder to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Influence of the starting vector Note that ∥ˆb∥ = 1, and suppose the eigenvectors have been normalized such that ∥vi∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Then, the coefficients {αi}i=1,n satisfy 1 = ∥α1v1 + · · · + αnvn∥ ≤ |α1| + · · · + |αn|, and let ξi = 1 |αi| n � j=1 |αj| − 1 ≥ 1 |αi| − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Suppose now that αj = δ for all j ̸= i, with δ small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Then, αi = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Under these conditions, the prefactor ξi is small, and the Arnoldi iteration exhibits a fast convergence for the ith eigenpair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Alternatively, if αi is very small, then ξi ≫ 1, and we have slow convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This observation gave rise to the rule-of-thumb that, for eigenvalue problems, the Arnoldi method will favor eigenvectors having large components in the starting vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Unless we have prior knowledge of the eigenvectors we wish to compute, starting the Arnoldi iteration with a genuinely random vector increases the probability that the eigenpairs of interest will converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 45 Figure 24: Convergence history of the leading eigenvalue as a function of the total integration time (defined as the product of the sampling period τ and the number m of Krylov iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=') for the flow past a circular cylinder at Re = 50 given different values of the sampling period τ: (a) the domain extends from −16 to +50 in the streamwise direction and (b) the domain extends from −16 to +100 in the streamwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='2 Influence of the domain extent for open shear flows We now verify the influence of the streamwise domain length on the convergence of the leading eigenpairs in open shear flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' We performed a series of computations for the flow past a circular cylinder at Re = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In this case, the leading eigenpair corresponds to the von K´arm´an vortex street mode associated with a supercritical Hopf bifurcation (as previously introduced in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Figure 24 depicts the evolution of the leading eigenvalue residual as a function of the total integration time T defined as the product of the number m of Krylov iterations and the sampling period τ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' T = mτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Letting Lx be the streamwise extent of the computational domain and U∞ the velocity scale, this figure highlights a rapid convergence of the leading eigenvalue once the total integration time exceeds mτ > Lx U∞ , with U∞ = 1 in the non-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This is easily explained by the fact that the usual choice is to start from an initial perturbation vector consisting of noise (which aims to excite the entire computational domain while minimizing any bias).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The leading eigenvalue then starts to converge after one flow-through time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 46 (a) mT = 50 10-1 T二2 T=1 10-4 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='25 10-7 E 10-10 10-13 I 10-16 50 100 150 200 250 300 (b) 102 mT = 100 T=2 10-1 T=1 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='5 10-4 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='25 10-7 C 10-10 10-13 10-16 50 100 150 200 250 300 Total integration time6 Conclusion Transition to turbulence is a long-standing problem in fluid dynamics, for which adopting a dy- namical system point of view has greatly increased our understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Specialized codes such as ChannelFlow [111, 112], OpenPipeFlow [273] or Semtex [33] are equipped with the set of tools nec- essary to probe the phase space of the Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Yet, by design, these are limited to canonical configurations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' plane Poiseuille and plane Couette flow for ChannelFlow) with infinite spans, and no such library is available for a general-purpose CFD code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This work therefore aims at providing a general-purpose introduction to the Krylov methods underlying numerous recent works on stability analysis of really high-dimensional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These methods are implemented in nekStab, an open-source and user-friendly toolbox to perform large-scale bifurcation analysis in Nek5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Using a time-stepper formulation and leveraging Krylov-based techniques, nekStab is capable of computing fixed points and periodic orbits, as well as computing the leading eigenpairs and singular triplets of the corresponding linearized Navier-Stokes operator to characterize their stability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The capabilities of nekStab and its underlying Krylov methods are showcased in a number of test cases available in the literature, including the canonical cylinder flow [19], the annular thermosyphon [157], the harmonically forced jet [238] and the flip-flop instability in the wake of side-by-side cylinders [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In all cases, excellent agreement has been obtained with the results available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' We believe that this bifurcation analysis plug-in for a code as established as Nek5000 could have far-reaching applications, both in academia and industry, and could improve our understanding of the transition to turbulence in non-canonical configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' By making the code open source, we hope to foster the ideas and efforts of our community as a whole and incorporate them into the development of a high-quality tool from which we can all benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Data availability The code and scripts presented in this review are freely available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='com/nekStab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Acknowledgment This work is built on the research of numerous former Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' students, including Fr´ed´eric Alizard [4], Stefania Cherubini [64], Jean-Christophe Loiseau [153], Alessandro Bucci [47], Mirko Farano [96], Francesco Picella [204] and Ricardo A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Frantz [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The authors acknowledge the support of GENCI under the following projects: A0072A06362/2020, A0092A06362/2021, and A0112A06362/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' We are also very grateful to Laurette Tuckerman for all the valuable discussions and insights that greatly improved this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' A nekStab, an open source toolbox for Nek5000 The numerical methods introduced in this paper are implemented and validated in a toolbox for linear stability analysis of steady and periodic flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' nekStab is an open-source toolbox that im- plements all the algorithms described in this article in less than 9000 lines of Fortran 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The toolbox uses the flexible and highly parallel data structure of Nek5000, which allows efficient com- putation of flows in complex geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' In the current implementation, when compiling Nek5000 nekStab is appended as a submodule so that the core of Nek5000 is only minimally modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 47 Nek5000, the workhorse of nekStab, is valued for its minimal dependencies and parallel perfor- mance, enabling computations on simple laptops up to high-performance computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Great care has been taken to limit the additional dependencies required to use nekStab, thanks to a non- invasive and self-contained design, no additional dependencies are introduced, and the need to modify the source code of Nek5000 has also been eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' To avoid compatibility issues and ease of use, nekStab interacts only with Nek5000 through the existing user interface, without re- quiring any changes to the original source code while retaining its native performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' nekStab directly uses Nek5000 time-stepper and closely adheres to its data structure, but only interacts with it through the existing user interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Given the design strategy for nekStab, potential conflicts with future updates to the main solver are unlikely, as the variables are separated by subroutines crafted to interface with Nek5000 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Like Nek5000, our toolbox also makes use of a few subroutines from the Linear Algebra PACKage (LAPACK) provided with nekStab to avoid exter- nal dependencies (version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='10 available at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='com/Reference-LAPACK included).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' All results presented in this work were computed with nekStab using the latest version of Nek5000 (commit 7ae03b1), available in github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='com/Nek5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The well-commented source code is maintained online in the public Git repository at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='com/nekStab, as is the collaborative online documentation at nekstab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='io with instructions and lightweight examples that can be quickly computed on a laptop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' These include canonical and more complex flows, such as flow past a cylinder or ad- jacent cylinders, a time-periodic axisymmetric jet, open and closed cavity flow, a stratified annular thermosyphon, flow past an airfoil, a backward step, sudden channel expansion, channel flow, and a Blasius boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The examples are compared to a reference case from the literature and aim to provide all the elements that new users might need to develop their own cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' To the best of our knowledge, nekStab is the first general-purpose computational framework capable of stabilizing fully 3D unstable periodic orbits (forced or unforced) and fixed points, as well as computing direct, adjoint modes and transient growth analyses for both steady and periodic flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The menu of options includes a matrix-free Newton GMRES solver as well as other classical techniques such as selective frequency damping (SFD) [1], BoostConv [70], Time-Delayed Feedback (TDF) [210, 237], and Dynamic Mode Tracking (DMT) [211], as well as post-processing routines such as the kinetic energy budget of the leading modes based on the Reynolds-Orr decomposi- tion [38, 154], steady-state base flow, and sensitivity analyzes [174].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' To facilitate adoption by the community (both academic and industrial), nekStab is released under the permissive BSD 3-Clause license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' All source code, examples, scripts, and tutorials can be viewed and downloaded for free at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='com/nekStab or nekstab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='io.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' We foresee more updates and future developments for nekStab on the horizon: integration of the algorithms proposed in [235] for the synthesis of linear optimal LQR con- trollers for large-scale systems, integration of the optimization algorithms in [102] and already implemented in Nek5000 by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Farano [97,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 98] for linear and nonlinear optimal perturbation analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' extension of the Newton-Krylov solver with pseudo-arclength continuation to compute branches of solutions even in the presence of folding points,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Computation of Lyapunov exponents using the algorithms presented in [27] and [104],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 48 B Proof of equation (57) Let us prove that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' given the Arnoldi factorization AQ = QH + βeT k qk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' the approximation of the ith eigenvector vi in the Krylov basis Q satisfies ���� Ä I − QQT ä vi ���� ≤ Ü n � j=1 j̸=i |αj| |αi| ê ε(k) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (58) Recall furthermore that the vector b used to seed this Krylov vector can be expressed as a linear combination of the eigenvectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' b = n � i=1 αivi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (59) The proof below follows closely the derivation given by Elias Jarlebring in his lecture notes at KTH (see https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='kth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='se/na/SF2524/matber15/arnoldiconv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='pdf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' It proceeds in three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Step 1 Consider an arbitrary vector u ∈ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Then the problem minimize z∈Ck ∥u − Qz∥2 (60) is a least-squares minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The matrix Q being orthonormal, its Moore-Penrose pseudoinverse Q† is equal to QT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Hence, the vector z solution to the minimization problem is simply given by z = QT u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' It implies in particular that min z∈Ck ∥u − Qz∥2 = ∥ Ä I − QQT ä u∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (61) Step 2 Our objective is to obtain an upper bound for the term ∥ �I − QQT � vi∥2, where vi is the ith eigenvector of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The proof is simplified by rescaling the right-hand side with αi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' ∥ Ä I − QQT ä αivi∥2 = min z∈Ck ∥αivi − Qz∥2 = min y∈Kk(A,b) ∥αivi − y∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (62) The Krylov subspace Kk(A, b) can be characterized with polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' The statement y ∈ Kk(A, b) is thus equivalent to the existence of a polynomial p ∈ Pk−1 such that y = p(A)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Hence ∥ Ä I − QQT ä αivi∥2 = min p∈Pk−1 ∥αivi − p(A)b∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (63) Step 3 The last step consists in inserting the expansion of b in terms of the eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' This leads to ∥ Ä I − QQT ä αivi∥2 = min p∈Pk−1 ∥αivi − p(A) n � j=1 αjvj∥2 = min p∈Pk−1 ∥αivi − n � j=1 αjp(µj)vj∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (64) 49 It can then easily be shown that the expression above is bounded from above by ∥ Ä I − QQT ä αivi∥2 ≤ Ü n � j=1 j̸=i |αj| ê ϵ(k) i , (65) where ϵ(k) i is given by ϵ(k) i = min p∈Pk−1 p(µi)=1 max j̸=i (|p(µj)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' (66) The proof of equation (57) is completed by dividing this upper bound by |αi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' ˚Akervik, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' Brandt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} +page_content=' 67' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FOT4oBgHgl3EQf1zQJ/content/2301.12940v1.pdf'} diff --git a/NtAyT4oBgHgl3EQftPkw/content/tmp_files/2301.00590v1.pdf.txt b/NtAyT4oBgHgl3EQftPkw/content/tmp_files/2301.00590v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..178fd04a83dde6f6cf6728229818c6830309f1b1 --- /dev/null +++ b/NtAyT4oBgHgl3EQftPkw/content/tmp_files/2301.00590v1.pdf.txt @@ -0,0 +1,539 @@ +arXiv:2301.00590v1 [astro-ph.IM] 2 Jan 2023 +End-to-end simulations of a near-infrared pyramid sensor on +Keck II +C. Planteta, G. Agapitoa, C. Giordanoa, S. Espositoa, P. Wizinowichb, and C. Bondc +aINAF - Osservatorio di Arcetri, 50125 Firenze, Italy +bW. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743, USA +cInstitute for Astronomy, University of Hawaii, 640 N. Aohoku Place, Hilo, HI 96720 +ABSTRACT +The future upgrade of Keck II telescope’s adaptive optics system will include a pyramid wavefront sensor working +in the near-infrared (J and H band). It will benefit from the recently developed avalanche photodiode arrays, +specifically the SAPHIRA (Selex) array, which provides a low noise (ă 1 e- at high frame rates). The system will +either work with a natural guide star (NGS) in a single conjugated adaptive optics system, or in a laser guide +star (LGS) mode. In this case, the pyramid would be used as a low-order sensor only. We report on a study +of the pyramid sensor’s performance via end-to-end simulations, applied to Keck’s specific case. We present the +expected Strehl ratio with optimized configurations in NGS mode, and the expected residual on low orders in +LGS mode. In the latter case, we also compare the pyramid to LIFT, a focal-plane sensor, demonstrating the +ability of LIFT to provide a gain of about 2 magnitudes for low-order sensing. +Keywords: Adaptive optics, Wavefront sensing, Infrared, Keck, Pyramid, LIFT +1. INTRODUCTION +The future upgrade of Keck II telescope’s Adaptive Optics (AO) system1 will include a pyramid wavefront sensor2 +working in the near-infrared (J and H band).3 The main goal of this upgrade is to perform direct imaging and +slit spectroscopy of exoplanets around M dwarfs. The flux from these stars is very faint at optical wavelengths, +but sufficient in the near-infrared to use as NGSs in a single conjugated AO system, given the adequate detector +technology. The recently developed avalanche photodiode arrays, such as the SAPHIRA (Selex), provide a low +noise (ă 1 e- at high frame rates) and are thus suitable for this application.4 In addition to the NGS mode, the +system will also provide a LGS mode. In this case, the pyramid would be used as a low-order sensor only. We +report on a study of the pyramid sensor’s performance via end-to-end simulations made with PASSATA.5 After +a quick summary of the simulation parameters (section 2), we present the expected Strehl ratio in NGS mode +(section 3), and the expected residual on low orders in LGS mode (section 4). In the latter case, the pyramid +will not benefit from a hardware rebinning of pixels, and thus will not be in a fully optimized configuration. For +this reason, we also compare the pyramid to LIFT,6 a focal-plane sensor, that could provide a better low-order +estimation at low flux. +2. SIMULATIONS PARAMETERS +We list in Table 1 the simulation parameters used for the different cases of this study. The chosen values for +the parameters that are optimized (modulation, frequency, control gain. . . ) are given in each specific case, and +we only state here the explored ranges of values. The wavefront modes (turbulent Karhunen-Lo`eve and Zernike) +are considered to be perfectly reproduced by the deformable mirror (DM). In all cases, the correction is made +with an integrator command, and the delay depends on the frequency, with the following rules (taken from ERIS +simulations experience7): +• f ą 666 Hz : 3 frames delay +Further author information: +C.P.: E-mail: plantet@arcetri.astro.it + +• 333 Hz ă f ď 666 Hz : 2 frames delay +• f ď 333 Hz : 1 frame delay +Table 1. Simulation parameters. +Parameter +NGS mode +LGS mode +20ˆ20 +32ˆ32 +20ˆ20 +32ˆ32 +Sensing band +1.5 µm - 1.8 µm (H band) +Pupil mask +Keck primary on 512 pixels +Keck primary on 256 pixels +Mode basis +250 KL modes +245 KL modes + 5 first Zernike +Total transmission (including QE) +0.3 +Sky background in H +14 mag/arcsec2 +Seeing +0.63” +Layers’ altitudes (km) +0, 500, 1000, 2000, 4000, 8000, 16000 +C2 +n profile (normalized in energy) +0.517, 0.119, 0.063, 0.061, 0.105, 0.081, 0.054 +Mean wind speed +9.5 m/s +Zenith angle +300 +Subaperture size +0.5625 m +0.35 m +0.5625 m +0.35 m +APD gain +30 +Excess noise factor +1.4 +Read-out noise +0.1 or 1 e´ +1 e´ +0.8 e´ +1 e´ +Dark current +0 or 100 e´/s +20 e´/s +100 e´/s +20 e´/s +Frequency range +300-1000 Hz +200-1000 Hz +Control gain range +0.1-0.6 +LIFT: 0.1-0.6 +LIFT: 0.1-0.6 +Pyramid: 0.15-10 +Pyramid:0.25-5 +Pyramid modulation radius range +1-3 λ{D +1-2 λ{D +0-2 λ{D +FoV +1” +Additional HO residual (non corrected) +60 nm +0 nm +60 nm +To these parameters, we add the following precisions for the LGS mode: +• High-order loop parameters: +– Sensor: SH 20ˆ20 with quad-cells estimating 250 modes. +– LGS = high flux point source at finite distance. +– Tip/tilt filtered and replaced by a residual jitter of 106 mas rms + turbulent tip/tilt. +– Control gain: 0.3. +• Focus loop (only in 32ˆ32 case): +– Correction frequency: 10 Hz. +– Input: focus residual from high-order control + sinusoid of period 5 seconds and amplitude 100 nm +(80 nm rms). +– Control gain range: 0.1-1 for LIFT, 0.1-4 for the pyramid. +Finally, for consistency with the error budget used in a previous study,3 we add a constant error of 165 nm +rms to the residual in NGS mode, representing miscellaneous errors from undetermined sources. + +3. NGS MODE +In this section, we study the performance of the pyramid, in terms of Strehl ratio, for different pupil samplings. +We first considered a pupil sampling of 20ˆ20 subapertures, in agreement with the current DM’s number of +actuators. However, the DM should be upgraded to a MEMS 32ˆ32. We thus study in a second step the impact +of increasing the pupil sampling to 32ˆ32, or to 40ˆ40 for robustness reasons. +3.1 Pyramid 20ˆ20 +We present here the results of the simulations with a pyramid 20ˆ20 in NGS mode (Fig. 3.1). The parameters, +listed in Table 2, are optimized in the ranges described previously to get the highest Strehl ratio. This optimiza- +tion is simply made by running simulations going through the whole set of parameters and selecting the best +one. +As we lacked information on the detector’s noise, we considered two cases: low noise (no dark current, read- +out noise = 0.1 e´) and high noise (dark current = 100 e´/s, read-out noise = 1 e´). The difference between +those two cases is not very significant (0.5 magnitude at faint end). +Table 2. Optimized parameters (high noise/low noise) for the pyramid 20ˆ20 in NGS mode. +Magnitude +8 +10 +12 +13 +14 +15 +Frequency (Hz) +1000/1000 +1000/1000 +1000/600 +1000/500 +600/300 +600/300 +Number of modes +250/250 +250/250 +170/152 +135/104 +65/54 +44/14 +Gain +0.3/0.3 +0.2/0.2 +0.15/0.25 +0.15/0.3 +0.3/0.55 +0.3/0.6 +Modulation radius (λ{D) +1.5 +1.5 +1.5 +1.5 +1.5 +2 +8 +9 +10 +11 +12 +13 +14 +15 +H-magnitude +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +SR +λ = 1265nm +λ = 1659nm +λ = 2145nm +λ = 2200nm +λ = 3805nm +λ = 4781nm +(a) Low noise +8 +9 +10 +11 +12 +13 +14 +15 +H-magnitude +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +SR +λ = 1265nm +λ = 1659nm +λ = 2145nm +λ = 2200nm +λ = 3805nm +λ = 4781nm +(b) High noise +8 +9 +10 +11 +12 +13 +14 +15 +H-Magnitude +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +SR(@K) +Dark = 0 -- RON = 0.1 +Dark = 100 -- RON = 1 +(c) Comparison +Figure 1. Strehl ratio as a function of H magnitude with a pyramid 20ˆ20 in NGS mode. Left: Low noise case. Middle: +High noise case. Right: Comparison of low and high noise cases in K band. +Overall, what we observe on the optimization of the parameters when we are going towards lower fluxes is: +• Decrease in frequency: to collect more flux and reduce the noise error. +• Increase the gain: we cannot remove the background in pyramid images, it is thus taken into account in +the normalization when computing the slopes. In the end, the slopes are proportional to the ratio star +flux/background, which decreases with respect to the magnitude. A higher gain is needed to compensate +that effect. The increase in gain is also needed at lower frequencies, as the correction is done less often. +• Increase in modulation: at low flux, the noise error makes the pyramid work in non-linear regime. The +modulation reduces the non-linearity error, at the price of a lower sensitivity (hence greater noise error). +A trade-off is made between those two errors to reach the lowest overall error. At high flux, using a high +modulation lowers the non-linearity error. + +• Decrease the number of modes: estimating less modes improves the noise propagation behavior at low +orders. +These results are consistent with the ones presented in an earlier study,3 with a difference of only a few +percents of Strehl ratio in K band. +3.2 Impact of a finer pupil sampling +Having a finer pupil sampling allows us to estimate more modes at high flux, but lowers the signal-to-noise ratio +(SNR) at low flux. We consider here only the impact at low flux, as it corresponds to more practical cases and +is more critical for the system design. +We simulated two different pupil samplings: 32ˆ32 and 40ˆ40 (in that case, only the subaperture size from +Table 1 is changed). The first one matches the MEMS mirror sampling, while the second would help calibrate +misregistration errors and thus gain in robustness. +The performance and optimized parameters at magnitude 14 are given in Table 3, for a dark current of 20 +e´/s and a read-out noise of 1 e´. The performance for the high noise case of the pyramid 20ˆ20 is recalled for +reference. It should be noted that the dark current does not have a significant impact here, the results can thus +be fairly compared. +Table 3. Optimized parameters and Strehl ratios for the pyramid 32ˆ32 and 40ˆ40 in NGS mode. +Frequency (Hz) +Number of modes +Gain +Modulation radius (λ{D) +Strehl ratio (K) +20ˆ20 +600 +65 +0.3 +1.5 +29.8% +32ˆ32 +200 +65 +0.75 +1.5 +26.7% +40ˆ40 +200 +65 +0.75 +1.5 +25.1% +The finer pupil sampling does not have a strong impact on performance: the loss of Strehl in Ks is 3% for +the 32ˆ32 and 5% for the 40ˆ40. Hence, it seems a reasonable choice to go towards a 40ˆ40 sampling, making +the system more reliable without a significant loss of performance at low flux. +4. LGS MODE +The goal of this section is to assess the achievable residual on tip/tilt and focus in LGS mode, for a NGS on +axis or at 15” off axis. We compare the pyramid to LIFT, in order to evaluate the gain of having a focal-plane +sensor for this low-order estimation. Indeed, as we cannot do a hardware rebin of pixels on the camera, the +pyramid would still utilise a fine sampling and would thus have poorer noise propagation properties for low-order +estimation than with a coarse sampling. +As in the previous section, we first considered a pyramid with 20ˆ20, and then checked the impact of a finer +sampling. For LIFT, the only design parameter that will have an impact on the performance is the pixel scale. +We consider here a pixel of 15 or 30 mas, corresponding respectively to a Nyquist and a Nyquist/2 sampling in +H band. +4.1 Pyramid 20ˆ20 +In this part, we only evaluate the residual on tip/tilt, as it is the most important feature of the low-order sensor. +The focus estimation will be included in the next section. For practical reasons, the number of reconstructed +modes for the pyramid is either 2 (lowest noise error) or 250 (lowest aliasing error). +We list in Tables 4 to 6 the optimized parameters for LIFT and the pyramid in each case, as well as the +residual on tip/tilt. The residuals obtained with LIFT and the pyramid are compared in Fig. 4.1. We find that +LIFT provides a gain of up to 2 magnitudes over the pyramid, either on axis or off axis. +The behavior of the optimized parameters for the pyramid is as described in section 3.1. In particular, we +can notice an increase in modulation at high flux when going off-axis: this is due to the increase in amplitude + +of high-order modes, for which the linearity must be improved. The flux is sufficiently high in that case to use +a strong modulation without a significant impact on noise error. +As concerns LIFT’s sampling, the pixel of 30 mas benefits from better noise propagation properties (better +SNR/pixel), but does not provide any significant improvement of the performance. On the contrary, it is less +efficient off axis, or at high flux in general. Indeed, the signal from high orders, normally far from the spot center, +gets more easily mixed with the low orders signal, which is within the spot center. This aliasing error is visible +at high flux, where the noise error is negligible, and gets higher when going off axis, where the Strehl ratio is +lower. The overall aliasing + noise error is in the end always better with the 15 mas pixel for the considered +magnitudes. +Table 4. Optimized parameters (on axis/off axis) for the pyramid 20ˆ20 in LGS mode. +Magnitude +10 +12 +13 +14 +15 +16 +Frequency (Hz) +1000/1000 +1000/1000 +1000/1000 +1000/1000 +200/200 +200/200 +Number of modes +250/250 +250/250 +2/250 +2/2 +2/2 +2/2 +Gain +0.5/0.25 +0.5/0.65 +0.75/0.65 +1/1 +3.5/3.5 +5/5 +Modulation radius (λ{D) +0/2 +0/0 +0/0 +0/0 +0/0 +0/1 +TT residual (nm rms) +21.4/48 +32.3/59.4 +44.6/73.3 +57.8/89.4 +88.6/123 +153.7/195.6 +Table 5. Optimized parameters (on axis/off axis) for LIFT with a 15 mas pixel in LGS mode. +Magnitude +10 +12 +13 +14 +15 +16 +Frequency (Hz) +1000/1000 +1000/1000 +1000/333 +333/333 +333/200 +200/200 +Gain +0.3/0.3 +0.3/0.3 +0.2/0.5 +0.4/0.5 +0.4/0.4 +0.4/0.3 +TT residual (nm rms) +18.9/42.3 +23.2/45.1 +27.3/48.5 +33.1/56.4 +42.9/67.5 +63.5/93.2 +Table 6. Optimized parameters (on axis/off axis) for LIFT with a 30 mas pixel in LGS mode. +Magnitude +10 +12 +13 +14 +15 +16 +Frequency (Hz) +1000/1000 +1000/1000 +1000/1000 +333/1000 +333/500 +200/333 +Gain +0.4/0.3 +0.3/0.2 +0.2/0.2 +0.5/0.2 +0.4/0.2 +0.4/0.3 +TT residual (nm rms) +25.7/62.5 +28.5/63.7 +31.5/66.5 +37.6/68.3 +45.6/81.1 +64.5/101.2 +(a) On axis +(b) 15” off axis +Figure 2. Comparison of the tip/tilt residual obtained with LIFT or the pyramid 20ˆ20 in LGS mode. Left: NGS on +axis. Right: NGS at 15” off axis. + +4.2 Finer pupil sampling +We now check the impact of having a finer-sampled pyramid on the performance at magnitude 14. We also verify +that we have a correct estimation of focus on both sensors, and we study the possibility of having 10 mas pixels +on LIFT (for design simplicity reasons). +The parameters and results are given in Table 7, for an optimization on tip/tilt correction only (the focus is +then added with the same loop parameters). +On axis, there is a clear advantage using LIFT for tip/tilt estimation, with a factor 2 in rms residual. The +estimation of focus does not affect the estimation of tip/tilt, whatever the sensor, and LIFT has a slight advantage +on this mode as well (factor 1.4). +Off axis, we still have a better estimation of tip/tilt with LIFT, but with less difference (factor 1.4 at best). +LIFT’s performance is actually weakly dependent on the frequency: at 1000 Hz, the tip/tilt residual is increased +by approximately 5 nm for the 15 mas and 10 mas pixels. The focus correction is similar in all cases, and the +tip/tilt estimation is always affected. This might be the result of the sensors getting less and less linear when +lowering the Strehl ratio (aliasing error discussed in the previous section). The effect seems stronger on LIFT at +fine samplings (15 and 10 mas), but for these cases, as well as the pyramid, it is equivalent to adding an error of +approximately 30-35 nm rms. For LIFT with 30 mas pixels, this error is lower, around 20 nm rms. This might +be due to the fact that the tip/tilt estimation alone was already affected by non-linear effects. +Table 7. Optimized parameters and residuals (on axis/off axis) for the pyramid 32ˆ32 and LIFT in LGS mode, with the +estimation of focus. +LIFT 30 mas +LIFT 15 mas +LIFT 10 mas +Pyramid +Frequency (Hz) +333/1000 +333/200 +333/333 +333/333 +Modulation radius (λ{D) +0/0 +Gain +0.4/0.1 +0.4/0.5 +0.4/0.3 +4/4 +TT residual (nm rms) +39.3/80.5 +34.3/67.5 +34.7/69.8 +68.9/95.9 +TT residual (with focus) +36.2/82.7 +33.6/75.9 +34.8/77.7 +69.3/100 +Gain on focus +0.6/0.4 +0.5/0.4 +0.7/0.3 +2/1.5 +Focus residual (nm rms) +37/52.9 +38.4/53.1 +35.9/53.2 +50.5/50.8 +5. CONCLUSION +We have studied the performance of a near-infrared pyramid for the next generation AO of Keck II, which will +include a classical AO mode (NGS mode) and a LGS mode. In NGS mode, the pyramid will provide a Strehl +ratio in K band of +37% at magnitude H = 14 and +80% at high flux (20ˆ20 configuration). The latter can +be increased with a finer pupil sampling (32ˆ32, or 40ˆ40) and a higher degree of correction (i. e. 32ˆ32 +DM), without degrading significantly the performance at low flux. The 40ˆ40 sampling would also provide more +robustness to errors such as misregistration. In LGS mode, the pyramid would not benefit from a hardware +rebin of pixels, and a focal plane sensor would be preferable to estimate low orders. We have demonstrated that +through a comparison with LIFT, which provides a gain of 2 magnitudes on tip/tilt up to 15” off axis and a +similar performance on focus. It was also shown that LIFT gives best results with images sampled at Nyquist +(15 mas pixels). In future works, we will explore more off-axis distances and seeing conditions to confirm the +advantage of using LIFT. We will also study the impact of the atmosphere dispersion on both sensors. +ACKNOWLEDGMENTS +This work was partly funded by INAF (Research Grant DD 27). The Keck II pyramid wavefront sensor is funded +by the National Science Foundation under Grant No. AST-1611623. + +REFERENCES +[1] Wizinowich, P., Le Mignant, D., Bouchez, A. H., Campbell, R. D., Chin, J. C., Contos, A. R., van Dam, +M. A., Hartman, S. K., Johansson, E. M., Lafon, R. E., et al., “The WM Keck Observatory laser guide star +adaptive optics system: overview,” Publications of the Astronomical Society of the Pacific 118(840), 297 +(2006). +[2] Ragazzoni, R., “Pupil plane wavefront sensing with an oscillating prism,” Journal of modern optics 43(2), +289–293 (1996). +[3] Wizinowich, P., Chun, M., Mawet, D., Agapito, G., Dekany, R., Esposito, S., Fusco, T., Guyon, O., Hall, D., +Plantet, C., and Rigaut, F., “Near-infrared wavefront sensing,” Proc.SPIE 9909, 9909 – 9909 – 13 (2016). +[4] Feautrier, P., Gach, J.-L., and Wizinowich, P., “State of the art IR cameras for wavefront sensing using +e-APD MCT arrays,” in [AO4ELT4 Proceedings], (2015). +[5] G. Agapito, A. Puglisi, S. E., “Passata: object oriented numerical simulation software for adaptive optics,” +Proc.SPIE 9909, 9909 – 9909 – 9 (2016). +[6] Meimon, S., Fusco, T., and Mugnier, L. M., “LIFT: a focal-plane wavefront sensor for real-time low-order +sensing on faint sources,” Optics letters 35(18), 3036–3038 (2010). +[7] Quir´os-Pacheco, F., Agapito, G., Riccardi, A., Esposito, S., Louarn, M. L., and Marchetti, E., “Performance +simulation of the eris pyramid wavefront sensor module in the vlt adaptive optics facility,” Proc.SPIE 8447, +8447 – 8447 – 12 (2012). + diff --git a/NtAyT4oBgHgl3EQftPkw/content/tmp_files/load_file.txt b/NtAyT4oBgHgl3EQftPkw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..47cb9929e0a8bfefd9fa4b262061ad20d16a7a70 --- /dev/null +++ b/NtAyT4oBgHgl3EQftPkw/content/tmp_files/load_file.txt @@ -0,0 +1,415 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf,len=414 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='00590v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='IM] 2 Jan 2023 End-to-end simulations of a near-infrared pyramid sensor on Keck II C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Planteta, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Agapitoa, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Giordanoa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Espositoa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Wizinowichb, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Bondc aINAF - Osservatorio di Arcetri, 50125 Firenze, Italy bW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Keck Observatory, 65-1120 Mamalahoa Hwy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=', Kamuela, HI 96743, USA cInstitute for Astronomy, University of Hawaii, 640 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Aohoku Place, Hilo, HI 96720 ABSTRACT The future upgrade of Keck II telescope’s adaptive optics system will include a pyramid wavefront sensor working in the near-infrared (J and H band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' It will benefit from the recently developed avalanche photodiode arrays, specifically the SAPHIRA (Selex) array, which provides a low noise (ă 1 e- at high frame rates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The system will either work with a natural guide star (NGS) in a single conjugated adaptive optics system, or in a laser guide star (LGS) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' In this case, the pyramid would be used as a low-order sensor only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' We report on a study of the pyramid sensor’s performance via end-to-end simulations, applied to Keck’s specific case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' We present the expected Strehl ratio with optimized configurations in NGS mode, and the expected residual on low orders in LGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' In the latter case, we also compare the pyramid to LIFT, a focal-plane sensor, demonstrating the ability of LIFT to provide a gain of about 2 magnitudes for low-order sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Keywords: Adaptive optics, Wavefront sensing, Infrared, Keck, Pyramid, LIFT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' INTRODUCTION The future upgrade of Keck II telescope’s Adaptive Optics (AO) system1 will include a pyramid wavefront sensor2 working in the near-infrared (J and H band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3 The main goal of this upgrade is to perform direct imaging and slit spectroscopy of exoplanets around M dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The flux from these stars is very faint at optical wavelengths, but sufficient in the near-infrared to use as NGSs in a single conjugated AO system, given the adequate detector technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The recently developed avalanche photodiode arrays, such as the SAPHIRA (Selex), provide a low noise (ă 1 e- at high frame rates) and are thus suitable for this application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4 In addition to the NGS mode, the system will also provide a LGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' In this case, the pyramid would be used as a low-order sensor only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' We report on a study of the pyramid sensor’s performance via end-to-end simulations made with PASSATA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 After a quick summary of the simulation parameters (section 2), we present the expected Strehl ratio in NGS mode (section 3), and the expected residual on low orders in LGS mode (section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' In the latter case, the pyramid will not benefit from a hardware rebinning of pixels, and thus will not be in a fully optimized configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' For this reason, we also compare the pyramid to LIFT,6 a focal-plane sensor, that could provide a better low-order estimation at low flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' SIMULATIONS PARAMETERS We list in Table 1 the simulation parameters used for the different cases of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The chosen values for the parameters that are optimized (modulation, frequency, control gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' ) are given in each specific case, and we only state here the explored ranges of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The wavefront modes (turbulent Karhunen-Lo`eve and Zernike) are considered to be perfectly reproduced by the deformable mirror (DM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' In all cases, the correction is made with an integrator command, and the delay depends on the frequency, with the following rules (taken from ERIS simulations experience7): f ą 666 Hz : 3 frames delay Further author information: C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' : E-mail: plantet@arcetri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='it 333 Hz ă f ď 666 Hz : 2 frames delay f ď 333 Hz : 1 frame delay Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Simulation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Parameter NGS mode LGS mode 20ˆ20 32ˆ32 20ˆ20 32ˆ32 Sensing band 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 µm - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='8 µm (H band) Pupil mask Keck primary on 512 pixels Keck primary on 256 pixels Mode basis 250 KL modes 245 KL modes + 5 first Zernike Total transmission (including QE) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3 Sky background in H 14 mag/arcsec2 Seeing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='63” Layers’ altitudes (km) 0, 500, 1000, 2000, 4000, 8000, 16000 C2 n profile (normalized in energy) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='517, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='119, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='063, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='061, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='105, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='081, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='054 Mean wind speed 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 m/s Zenith angle 300 Subaperture size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5625 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='35 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5625 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='35 m APD gain 30 Excess noise factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4 Read-out noise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1 or 1 e´ 1 e´ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='8 e´ 1 e´ Dark current 0 or 100 e´/s 20 e´/s 100 e´/s 20 e´/s Frequency range 300-1000 Hz 200-1000 Hz Control gain range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='6 LIFT: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='6 LIFT: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='6 Pyramid: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='15-10 Pyramid:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='25-5 Pyramid modulation radius range 1-3 λ{D 1-2 λ{D 0-2 λ{D FoV 1” Additional HO residual (non corrected) 60 nm 0 nm 60 nm To these parameters, we add the following precisions for the LGS mode: High-order loop parameters: – Sensor: SH 20ˆ20 with quad-cells estimating 250 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' – LGS = high flux point source at finite distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' – Tip/tilt filtered and replaced by a residual jitter of 106 mas rms + turbulent tip/tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' – Control gain: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Focus loop (only in 32ˆ32 case): – Correction frequency: 10 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' – Input: focus residual from high-order control + sinusoid of period 5 seconds and amplitude 100 nm (80 nm rms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' – Control gain range: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1-1 for LIFT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1-4 for the pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Finally, for consistency with the error budget used in a previous study,3 we add a constant error of 165 nm rms to the residual in NGS mode, representing miscellaneous errors from undetermined sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' NGS MODE In this section, we study the performance of the pyramid, in terms of Strehl ratio, for different pupil samplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' We first considered a pupil sampling of 20ˆ20 subapertures, in agreement with the current DM’s number of actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' However, the DM should be upgraded to a MEMS 32ˆ32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' We thus study in a second step the impact of increasing the pupil sampling to 32ˆ32, or to 40ˆ40 for robustness reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1 Pyramid 20ˆ20 We present here the results of the simulations with a pyramid 20ˆ20 in NGS mode (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The parameters, listed in Table 2, are optimized in the ranges described previously to get the highest Strehl ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' This optimiza- tion is simply made by running simulations going through the whole set of parameters and selecting the best one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' As we lacked information on the detector’s noise, we considered two cases: low noise (no dark current, read- out noise = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1 e´) and high noise (dark current = 100 e´/s, read-out noise = 1 e´).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The difference between those two cases is not very significant (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 magnitude at faint end).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Optimized parameters (high noise/low noise) for the pyramid 20ˆ20 in NGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Magnitude 8 10 12 13 14 15 Frequency (Hz) 1000/1000 1000/1000 1000/600 1000/500 600/300 600/300 Number of modes 250/250 250/250 170/152 135/104 65/54 44/14 Gain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='2/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='15/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='15/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='6 Modulation radius (λ{D) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 2 8 9 10 11 12 13 14 15 H-magnitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='0 SR λ = 1265nm λ = 1659nm λ = 2145nm λ = 2200nm λ = 3805nm λ = 4781nm (a) Low noise 8 9 10 11 12 13 14 15 H-magnitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='0 SR λ = 1265nm λ = 1659nm λ = 2145nm λ = 2200nm λ = 3805nm λ = 4781nm (b) High noise 8 9 10 11 12 13 14 15 H-Magnitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='0 SR(@K) Dark = 0 -- RON = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1 Dark = 100 -- RON = 1 (c) Comparison Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Strehl ratio as a function of H magnitude with a pyramid 20ˆ20 in NGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Left: Low noise case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Middle: High noise case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Right: Comparison of low and high noise cases in K band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Overall, what we observe on the optimization of the parameters when we are going towards lower fluxes is: Decrease in frequency: to collect more flux and reduce the noise error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Increase the gain: we cannot remove the background in pyramid images, it is thus taken into account in the normalization when computing the slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' In the end, the slopes are proportional to the ratio star flux/background, which decreases with respect to the magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' A higher gain is needed to compensate that effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The increase in gain is also needed at lower frequencies, as the correction is done less often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Increase in modulation: at low flux, the noise error makes the pyramid work in non-linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The modulation reduces the non-linearity error, at the price of a lower sensitivity (hence greater noise error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' A trade-off is made between those two errors to reach the lowest overall error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' At high flux, using a high modulation lowers the non-linearity error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Decrease the number of modes: estimating less modes improves the noise propagation behavior at low orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' These results are consistent with the ones presented in an earlier study,3 with a difference of only a few percents of Strehl ratio in K band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='2 Impact of a finer pupil sampling Having a finer pupil sampling allows us to estimate more modes at high flux, but lowers the signal-to-noise ratio (SNR) at low flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' We consider here only the impact at low flux, as it corresponds to more practical cases and is more critical for the system design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' We simulated two different pupil samplings: 32ˆ32 and 40ˆ40 (in that case, only the subaperture size from Table 1 is changed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The first one matches the MEMS mirror sampling, while the second would help calibrate misregistration errors and thus gain in robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The performance and optimized parameters at magnitude 14 are given in Table 3, for a dark current of 20 e´/s and a read-out noise of 1 e´.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The performance for the high noise case of the pyramid 20ˆ20 is recalled for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' It should be noted that the dark current does not have a significant impact here, the results can thus be fairly compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Optimized parameters and Strehl ratios for the pyramid 32ˆ32 and 40ˆ40 in NGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Frequency (Hz) Number of modes Gain Modulation radius (λ{D) Strehl ratio (K) 20ˆ20 600 65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='8% 32ˆ32 200 65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='7% 40ˆ40 200 65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1% The finer pupil sampling does not have a strong impact on performance: the loss of Strehl in Ks is 3% for the 32ˆ32 and 5% for the 40ˆ40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Hence, it seems a reasonable choice to go towards a 40ˆ40 sampling, making the system more reliable without a significant loss of performance at low flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' LGS MODE The goal of this section is to assess the achievable residual on tip/tilt and focus in LGS mode, for a NGS on axis or at 15” off axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' We compare the pyramid to LIFT, in order to evaluate the gain of having a focal-plane sensor for this low-order estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Indeed, as we cannot do a hardware rebin of pixels on the camera, the pyramid would still utilise a fine sampling and would thus have poorer noise propagation properties for low-order estimation than with a coarse sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' As in the previous section, we first considered a pyramid with 20ˆ20, and then checked the impact of a finer sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' For LIFT, the only design parameter that will have an impact on the performance is the pixel scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' We consider here a pixel of 15 or 30 mas, corresponding respectively to a Nyquist and a Nyquist/2 sampling in H band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1 Pyramid 20ˆ20 In this part, we only evaluate the residual on tip/tilt, as it is the most important feature of the low-order sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The focus estimation will be included in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' For practical reasons, the number of reconstructed modes for the pyramid is either 2 (lowest noise error) or 250 (lowest aliasing error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' We list in Tables 4 to 6 the optimized parameters for LIFT and the pyramid in each case, as well as the residual on tip/tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The residuals obtained with LIFT and the pyramid are compared in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' We find that LIFT provides a gain of up to 2 magnitudes over the pyramid, either on axis or off axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The behavior of the optimized parameters for the pyramid is as described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' In particular, we can notice an increase in modulation at high flux when going off-axis: this is due to the increase in amplitude of high-order modes, for which the linearity must be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The flux is sufficiently high in that case to use a strong modulation without a significant impact on noise error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' As concerns LIFT’s sampling, the pixel of 30 mas benefits from better noise propagation properties (better SNR/pixel), but does not provide any significant improvement of the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' On the contrary, it is less efficient off axis, or at high flux in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Indeed, the signal from high orders, normally far from the spot center, gets more easily mixed with the low orders signal, which is within the spot center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' This aliasing error is visible at high flux, where the noise error is negligible, and gets higher when going off axis, where the Strehl ratio is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The overall aliasing + noise error is in the end always better with the 15 mas pixel for the considered magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Optimized parameters (on axis/off axis) for the pyramid 20ˆ20 in LGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Magnitude 10 12 13 14 15 16 Frequency (Hz) 1000/1000 1000/1000 1000/1000 1000/1000 200/200 200/200 Number of modes 250/250 250/250 2/250 2/2 2/2 2/2 Gain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='75/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='65 1/1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 5/5 Modulation radius (λ{D) 0/2 0/0 0/0 0/0 0/0 0/1 TT residual (nm rms) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4/48 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3/59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='6/73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='8/89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='6/123 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='7/195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='6 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Optimized parameters (on axis/off axis) for LIFT with a 15 mas pixel in LGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Magnitude 10 12 13 14 15 16 Frequency (Hz) 1000/1000 1000/1000 1000/333 333/333 333/200 200/200 Gain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='2/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3 TT residual (nm rms) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='9/42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='2/45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3/48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1/56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='9/67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5/93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='2 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Optimized parameters (on axis/off axis) for LIFT with a 30 mas pixel in LGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Magnitude 10 12 13 14 15 16 Frequency (Hz) 1000/1000 1000/1000 1000/1000 333/1000 333/500 200/333 Gain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='2/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3 TT residual (nm rms) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='7/62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5/63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='7 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5/66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='6/68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='6/81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5/101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='2 (a) On axis (b) 15” off axis Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Comparison of the tip/tilt residual obtained with LIFT or the pyramid 20ˆ20 in LGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Left: NGS on axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Right: NGS at 15” off axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='2 Finer pupil sampling We now check the impact of having a finer-sampled pyramid on the performance at magnitude 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' We also verify that we have a correct estimation of focus on both sensors, and we study the possibility of having 10 mas pixels on LIFT (for design simplicity reasons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The parameters and results are given in Table 7, for an optimization on tip/tilt correction only (the focus is then added with the same loop parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' On axis, there is a clear advantage using LIFT for tip/tilt estimation, with a factor 2 in rms residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The estimation of focus does not affect the estimation of tip/tilt, whatever the sensor, and LIFT has a slight advantage on this mode as well (factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Off axis, we still have a better estimation of tip/tilt with LIFT, but with less difference (factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4 at best).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' LIFT’s performance is actually weakly dependent on the frequency: at 1000 Hz, the tip/tilt residual is increased by approximately 5 nm for the 15 mas and 10 mas pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The focus correction is similar in all cases, and the tip/tilt estimation is always affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' This might be the result of the sensors getting less and less linear when lowering the Strehl ratio (aliasing error discussed in the previous section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The effect seems stronger on LIFT at fine samplings (15 and 10 mas), but for these cases, as well as the pyramid, it is equivalent to adding an error of approximately 30-35 nm rms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' For LIFT with 30 mas pixels, this error is lower, around 20 nm rms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' This might be due to the fact that the tip/tilt estimation alone was already affected by non-linear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' Optimized parameters and residuals (on axis/off axis) for the pyramid 32ˆ32 and LIFT in LGS mode, with the estimation of focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' LIFT 30 mas LIFT 15 mas LIFT 10 mas Pyramid Frequency (Hz) 333/1000 333/200 333/333 333/333 Modulation radius (λ{D) 0/0 Gain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3 4/4 TT residual (nm rms) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3/80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3/67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='7/69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='8 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='9/95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='9 TT residual (with focus) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='2/82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='7 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='6/75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='9 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='8/77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='7 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3/100 Gain on focus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='6/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='7/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='3 2/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5 Focus residual (nm rms) 37/52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='9 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='4/53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='1 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='9/53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='5/50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' CONCLUSION We have studied the performance of a near-infrared pyramid for the next generation AO of Keck II, which will include a classical AO mode (NGS mode) and a LGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' In NGS mode, the pyramid will provide a Strehl ratio in K band of 37% at magnitude H = 14 and 80% at high flux (20ˆ20 configuration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The latter can be increased with a finer pupil sampling (32ˆ32, or 40ˆ40) and a higher degree of correction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' 32ˆ32 DM), without degrading significantly the performance at low flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The 40ˆ40 sampling would also provide more robustness to errors such as misregistration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' In LGS mode, the pyramid would not benefit from a hardware rebin of pixels, and a focal plane sensor would be preferable to estimate low orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' We have demonstrated that through a comparison with LIFT, which provides a gain of 2 magnitudes on tip/tilt up to 15” off axis and a similar performance on focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' It was also shown that LIFT gives best results with images sampled at Nyquist (15 mas pixels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' In future works, we will explore more off-axis distances and seeing conditions to confirm the advantage of using LIFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' We will also study the impact of the atmosphere dispersion on both sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was partly funded by INAF (Research Grant DD 27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'} +page_content=' The Keck II pyramid wavefront sensor is funded by the National Science Foundation under Grant No.' metadata={'source': 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a/ONE0T4oBgHgl3EQf0gLr/content/tmp_files/2301.02688v1.pdf.txt b/ONE0T4oBgHgl3EQf0gLr/content/tmp_files/2301.02688v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f55135b5581640cc61fef73079e692ff4eb9f86a --- /dev/null +++ b/ONE0T4oBgHgl3EQf0gLr/content/tmp_files/2301.02688v1.pdf.txt @@ -0,0 +1,406 @@ +arXiv:2301.02688v1 [math.CO] 6 Jan 2023 +NORMALLY LOCATED POLYHEDRA +IVAN ARZHANTSEV +Abstract. Lattice polyhedra Q1 and Q2 with the same tail cone are said to be normally +located if every lattice point in the Minkowski sum Q1 + Q2 is the sum of lattice points +from Q1 and Q2, respectively. We prove that if the normal fan of Q1 refines the normal fan +of Q2, then there is a positive integer k such that for any positive integer s the polyhedra +skQ1 and skQ2 are normally located. This result is based on an interpretation of the +problem in terms of graded algebras and earlier results on surjectivity of the multiplicaiton +map on homogeneous components. Also we provide an example of two lattice triangles P +and Q on the plane such that for any positive integer k the triangles kP and kQ are not +normally located. +1. Introduction +Let us consider the lattice Zd and the rational vector space Qd generated by Zd. By a +lattice polytope we mean a convex polytope P in Qd with vertices in Zd. Let us assume +additionally that the lattice points in P generate the lattice Zd; this can be achieved by +replacing Zd with a proper sublattice. +It is easy to construct examples of two lattice polytopes P and Q such that the Minkowski +sum P + Q contains a lattice point that is not a sum of lattice points from P and Q, +respective; see, e.g., [8]. Moreover, starting from dimension 3 it may happen even when +P = Q. +Definition 1. A lattice polytope P is called normal if for every positive integer s and every +lattice point z ∈ sP there are lattice points z1, . . . , zs ∈ P such that z = z1 + . . . + zs. +Normal polytopes play an important role in many areas of modern mathematics, see [5] +for a recent survey on this subject. In particular, such polytopes define integrally closed +graded monoid algebras and projectively normal embeddings of projective toric varieties. +Let us recall that a lattice polytope P is smooth if the primitive edge vectors at every +vertex of P form a basis of Zd. Smooth polytopes correspond to projective embeddings of +smooth toric varieties. Oda’s question [5, Section 1] asks whether every smooth polytope +is normal. This question is open in all dimensions ⩾ 3. +The proof of the following theorem may be found in [3, Proposition 1.3.3] or [4]. +Theorem 1. For every lattice polytope P in Qd the polytope (d − 1)P is normal. +In particular, in dimension 2 any lattice polygon is normal. +The aim of this paper is to generalize the property of normality to a pair of polytopes. +Definition 2. Lattice polytopes P and Q in Qd are said to be normally located if for every +lattice point z ∈ P + Q there are lattice points z′ ∈ P and z′′ ∈ Q such that z = z′ + z′′. +2010 Mathematics Subject Classification. Primary 11P21, 52B20; Secondary 14M25, 52B11. +Key words and phrases. Polytope, polyhedron, lattice, Minkowski sum, graded algebra, toric variety. +1 + +2 +IVAN ARZHANTSEV +Oda’s Conjecture (see [8] or [6, Section 1]) states the following. Let P and Q be lattice +polytopes. Assume that P is smooth and the normal fan of Q refines that of P. Then +the polytopes P and Q are normally located. This is a generalization of Oda’s question on +normality of a smooth polytope. +In [6, Theorem 1.1], it is shown that if P and Q are lattice polygons such that the normal +fan of Q refines that of P, then P and Q are normally located. In this paper we prove +a version of Oda’s Conjecture (Theorem 3) which generalizes [6, Theorem 1.1] to higher +dimensions. The result is obtained not only for polytopes, but also for polyhedra with the +same tail cone. +The paper is organized as follows. In Section 2 we give an explicit example of two lattice +triangles P and Q on the plane such that for any positive integer k the triangles kP and +kQ are not normally located. This example is intended to demonstrate that, unlike the +normality property, the property of normal location cannot always be achieved just by +rescaling two given polytopes. +In Section 3 we recall basic definitions and facts on polyhedra, their Minkowski sums +and normal fans. Also we consider polyhedra that appear as fibers of a projection of the +positive octant in a bigger lattice to a smaller lattice. Section 4 is devoted to interpretations +of the objects defined above in terms of graded algebras and the multiplication map on +homogeneous components. We recall the results of [1] that allow to relate the property +of normal location (up to scalar) of a pair of polyhedra in fibers over two points with the +location of these points with respect to the so-called GIT-fan. Finally, in Section 5 we +explain how to realize a pair of polyhedra via the fiber construction and prove our main +result (Theorem 3). +2. An example in dimension 2 +Let us show that, in contrast to Theorem 1, for two lattice triangles absence of the +property of normal location is not just a question of scale. +Proposition 1. There are two lattice triangles P and Q in Q2 such that for every positive +integer k the triangles kP and kQ are not normally located. +Proof. Take P = Conv(a1, a2, a3) and Q = Conv(b1, b2, b3), where +a1 = (165, 0), a2 = (175, 0), a3 = (0, 385) +and +b1 = (0, 0), b2 = (35, 0), b3 = (0, 77). +Equivalently, the triangle P is given by inequalities +7x + 3y ⩾ 1155, +11x + 5y ⩽ 1925, +y ⩾ 0 +and Q is given by +11x + 5y ⩽ 385, +x ⩾ 0, +y ⩾ 0. +The Minkowski sum P + Q is Conv(c1, c2, c3, c4), where +c1 = (165, 0), +c2 = (210, 0), +c3 = (0, 385), +c4 = (0, 462). +The inequalites that determine P + Q are +7x + 3y ⩾ 1155, +11x + 5y ⩽ 2310, +x ⩾ 0, +y ⩾ 0. +It is easy to check that for every positive integer k the point s = (1, 385k − 2) is contained +in kP + kQ. Assume that s = p + q, where p ∈ kP ∩ Z2 and q ∈ kQ ∩ Z2. Then only two +cases are possible. + +NORMALLY LOCATED POLYHEDRA +3 +Case 1. Let p = (1, a) and q = (0, 385k − 2 − a) for some non-negative integer a. Since +p is contained in P, we have +7 + 3a ⩾ 1155k +and +11 + 5a ⩽ 1925k. +These inequalities may be rewritten as +a ⩾ 385k − 2 − 1 +3 +and +a ⩽ 385k − 2 − 1 +5. +Since a is integer, we come to a contradiction. +Case 2. Let p = (0, a) and q = (1, 385k − 2 − a) for some non-negative integer a. Since +p is contained in P, we have +3a ⩾ 1155k +and +5a ⩽ 1925k, +so +a ⩾ 385k +and +a ⩽ 385k. +We conclude that a = 385k. Since q lies in Q, we have +385k − 2 − a = −2 > 0. +These two contradictions complete the proof. +□ +3. Generalities on polyhedra +By a polyhedron in Qd we mean the intersection of finitely many closed affine half spaces. +A lattice polyhedron is a polyhedron in Qd whose vertices are in Zd. Note that a polytope +can be defined as a bounded polyhedron in Qd. For a polyhedron Q we define its relative +interior as the set obtained by removing all proper faces from Q. Let us denote the relative +interior of Q by Q◦. Further, a cone in Qd is the intersection of finitely many closed linear +half spaces. A cone is pointed if it contains no line. If a cone has dimension at least 2, it is +pointed if and only if it is generated by its one-dimensional faces. +The set of all polyhedra in Qd comes with a natural structure of a commutative semigroup: +one defines the Minkowski sum of two polyhedra Q1 and Q2 to be the polyhedron +Q1 + Q2 := {w1 + w2; w1 ∈ Q1, w2 ∈ Q2}. +In the same way one may define the Minkowski sum of two arbitrary subsets in Qd. +Any polyhedron allows a Minkowski sum decomposition Q = P +σ, where P is a polytope +and σ is a cone in Qd. In this decomposition, the tail cone σ is unique; it is given by +σ = {w ∈ Qd; w′ + tw ∈ Q for all w′ ∈ Q, t ∈ Q⩾0}. +A polyhedron with the tail cone σ is called a σ-polyhedron. For example, polytopes are +precisely σ-polyhedra with σ = {0}. +It is easy to check that for a fixed cone σ the set of all σ-polyhedra forms a commutative +semigroup with respect to the Minkowski sum. +Let us recall that with any σ-polyhedron Q in Qd one may associate the normal fan N (Q): +any point v ∈ Q defines the cone τ(v) consisting of all linear functions on Qd which reach +their maximal value on Q at the point v. The collection of cones N (Q) = {τ(v) | v ∈ Q} is +finite and it is a fan in a sense that a face of any cone in N (Q) is contained in N (Q) and +the intersection of any two cones in N (Q) is a face of each of them. Moreover, all cones in +N (Q) are pointed if and only if Q has full dimension in Qd. + +4 +IVAN ARZHANTSEV +The support of a fan N is the union of all cones in N . The support of the normal fan +N (Q) equals the dual cone +σ∨ := {l ∈ (Qd)∗ | l|σ ⩾ 0}. +We say that a fan N1 refines a fan N2, if every cone in N1 is contained in some cone +in N2. Let N1 and N2 be two fans with the same support. The coarsest common refinement +of N1 and N2 is the fan N with the same support, whose cones are τ1 ∩ τ2, where τ1 ∈ N1 +and τ2 ∈ N2. +It is well-known that for normal fans N (Q1) and N (Q2) of two σ-polyhedra Q1 and Q2 +the coarsest common refinement N coincides with the normal fan N (Q1 + Q2). +Now let us consider a surjective homomorphism of lattices π: Zn → Zm and the induced +linear map of vector spaces π: Qn → Qm. We denote by Qn +⩾0 the cone of vectors in Qn with +non-negative coordinates and let C := π(Qn +⩾0) ⊆ Qm. Consider the cone σ := π−1(0)∩Qn +⩾0. +With any point u ∈ C one associates the polyhedron P(u) := π−1(u) ∩ Qn +⩾0. +Lemma 1. For any u ∈ C the polyhedron P(u) is a σ-polyhedron. +Proof. For a vector w ∈ Qn, the condition w′ + tw ∈ P(u) for all w′ ∈ P(u) and all t ∈ Q⩾0 +means that w ∈ Qn +⩾0 and u = π(w′ + tw) = π(w′) + π(tw) = u + tπ(w). It is equivalent to +w ∈ Qn +⩾0 and π(w) = 0, or w ∈ σ. +□ +Clearly, for every u ∈ C there is a positive integer r such that P(ru) is a lattice poly- +hedron. Also it is easy to check that P(u1) + P(u2) is contained in P(u1 + u2) for all +u1, u2 ∈ C. +We are interested in the following three properties of a pair (u1, u2) with u1, u2 ∈ C: +(P1) +P(u1) + P(u2) = P(u1 + u2); +(P2) +(P(u1) ∩ Zn) + (P(u2) ∩ Zn) = P(u1 + u2) ∩ Zn; +(P3) +there exists k ∈ Z>0 such that for any s ∈ Z>0 we have +(P(sku1) ∩ Zn) + (P(sku2) ∩ Zn) = P(sk(u1 + u2)) ∩ Zn. +4. Graded algebras +In this section we introduce an algebraic interpretation of the objects discussed above. +The projection π: Zn → Zm gives rise to an effective Zm-grading on the polynomial algebra +A := K[x1, . . . , xn]. Namely, we put deg(xi) = π(ei), where e1, . . . , en is the standard basis +of the lattice Zn. For further purposes we assume the ground field K to be an algebraically +closed field of characteristic zero. +Remark 1. The Linearization Problem [7] claims that up to automorphism any effective +Zm-grading on K[x1, . . . , xn] is obtained this way. +Below we follow the presentation given in [1] in a somewhat more general situation. Let +A be an associative, commutative, integral, finitely generated algebra with unit over K. +Suppose that A is graded by the lattice Zm, i.e., we have +A += +� +u∈Zm +Au +with +Au1 · Au2 ⊆ Au1+u2. + +NORMALLY LOCATED POLYHEDRA +5 +By the weight cone of A we mean the cone C(A) ⊆ Qm generated by all u ∈ Zm with +Au ̸= 0. We investigate the following problem: given u1, u2 ∈ C(A) ∩ Zm, does there exist +an integer k > 0 such that for any s > 0 the multiplication map +Asku1 ⊗K Asku2 → Ask(u1+u2), +f ⊗ g �→ fg +is surjective? We call a pair u1, u2 ∈ C(A) ∩ Zm generating if it has this property. If A is +a polynomial algebra with Zm-grading given by the projection π, this is precisely property +(P3) for a pair (u1, u2) of lattice points in C. +Let us recall from [2] the concept of the GIT-fan associated to a graded algebra. A Zm- +grading on A defines an action of the torus T := Spec(K[Zm]) on X := Spec(A) such that +for any u ∈ Zm, the elements f ∈ Au are precisely the semiinvariants of the character +χu : T → K∗, i.e., each f ∈ Au satisfies +f(t·x) +:= +χu(t)f(x). +The orbit cone of a (closed) point x ∈ X is the cone ω(x) ⊆ Qm generated by all u ∈ C(A) +admitting an f ∈ Au with f(x) ̸= 0. The collection of orbit cones is finite, and thus one +may associate to any element u ∈ C(A) its GIT-cone: +λ(u) +:= +� +x∈X, +u∈ω(x) +ω(x). +These GIT-cones cover the weight cone C(A) and, by [2, Theorem 3.11], the collection Λ(A) +of all GIT-cones is a fan in the sense that if λ ∈ Λ(A) then also every face of λ belongs to +Λ(A), and for τ, λ ∈ Λ(A) the intersection τ ∩ λ is a face of both λ and τ. Note that we +allow here a fan to have cones containing lines. +Theorem 2. [1, Theorem 1.1] +(1) If u1, u2 ∈ C(A) ∩ Zm is a generating pair, then the weights u1, u2 lie in a common +GIT-cone λ ∈ Λ(A). +(2) If u1, u2 ∈ C(A) ∩ Zm lie in a common GIT-cone λ ∈ Λ(A) and u1 belongs to the +relative interior λ◦, then (u1, u2) is a generating pair. +If two weights u1, u2 ∈ C(A) ∩ Zm lie on the boundary of a common GIT-cone λ ∈ Λ(A), +then no general statement in terms of the GIT-fan is possible: it may happen that u1, u2 +is generating, and also it may happen that u1, u2 is not generating. For the first case there +are obvious examples, and for the latter we present the following one. +Example 1. [1, Example 1.2] Consider the polynomial ring A := K[x1, x2, x3, x4]. Then +one may define a Z2-grading on A by setting +deg(x1) := (4, 1), +deg(x2) := (2, 1), +deg(x3) := (1, 2), +deg(x4) := (1, 3). +The pair u1 := (2, 1) and u2 := (1, 2) is contained in a common GIT-cone but it is not +generating: one checks directly that the monomials x1xs−2 +2 +xs−1 +3 +x4 ∈ As(u1+u2) can never be +obtained by multiplying elements from Asu1 and Asu2. +Remark 2. In [1, Theorem 1.5], a criterion for a pair of weights (u1, u2) in one GIT-cone to +be generating is given in terms of normality of the image of a morphism between certain +quotient spaces. +The proof of [1, Theorem 1.1] is based on several propositions. A modification of one of +them will be used below. To formulate this modification, we need some more notions from +the theory of graded algebras. + +6 +IVAN ARZHANTSEV +A subalgebra B of a graded algebra A = ⊕u∈ZmAu is homogeneous if B is the direct sum +of intersections of B with homogeneous components of A. Every homogeneous subalgebra +in A inherits a Zm-grading. +With any weight u ∈ Zm one associates a homogeneous subalgebra A(u) := ⊕r⩾0Aru. +Note that the subalgebra A(u) is Z⩾0-graded. +Further, with any weights u1 and u2 one associates a homogeneous subalgebra A(u1, u2) +in A(u1 + u2) defined as +A(u1, u2) := +� +r⩾0 +Aru1 · Aru2. +We say that a homogeneous subalgebra B of a Z⩾0-graded algebra A is big, if the radical +of the ideal B+ := ⊕r>0Br coincides with A+ := ⊕r>0Ar. +Let A be the polynomial algebra K[x1, . . . , xn] with Zm-grading given by a projection +π: Zn → Zm. For every v ∈ Zn +⩾0 we denote by xv the monomial xv1 +1 . . . xvn +n . +Proposition 2. Condition (P1) on a pair of weights (u1, u2) is equivalent to each of the +conditions: +(1) the weights u1, u2 lie in a common GIT-cone; +(2) the subalgebra A(u1, u2) is big in A(u1 + u2). +Proof. The equivalence of conditions (1) and (2) is proved in [1, Proposition 2.1]. +Let us prove that (P1) implies (2). Assume that P(u1 + u2) = P(u1) + P(u2) and take a +monomial xv ∈ A(u1 +u2). We have to prove that xv is contained in the radical of the ideal +A(u1, u2)+, i.e., there is a positive integer q such that xqv ∈ A(u1, u2)+. By assumption, +we have v = v′ + v′′ with v′ ∈ P(u1) and v′′ ∈ P(u2). Then there is q ∈ Z>0 such that +qv = qv′ + qv′′ with qv′ ∈ P(u1) ∩ Zn and qv′′ ∈ P(u2) ∩ Zn. This proves the assertion. +Now we come to implication (2) ⇒ (P1). Note that v ∈ P(u1 + u2) if and only if there +is s ∈ Z>0 such that sv ∈ P(s(u1 + u2)) ∩ Zn or, equivalently, the monomial xsv lies in +A(u1 + u2)+. By assumption, this implies that xsv is contained in the radical of A(u1, u2)+, +or there exists t ∈ Z>0 such that xtsv ∈ A(u1, u2)+. The later condition means that there +are v′ ∈ P(tsu1)∩Zn and v′′ ∈ P(tsu2)∩Zn such that tsv = v′ +v′′. This condition implies +v ∈ P(u1) + P(u2). +□ +5. Positive results on normal location +We keep the notation introduced in the previous section. In particular, every surjec- +tive homomorphism of lattices π: Zn → Zm gives rise to a Zm-grading on the algebra +K[x1, . . . , xn], and we speak about the weight cone, the orbit cones and the GIT-cones cor- +responding to this grading. In this situation, the weight cone coincides with C = π(Qn +⩾0) +and the orbits cones are precisely the cones generated by all subsets of the set {w1, . . . , wn}, +where wi := π(ei). +Let σ be a pointed cone in Qd. Taking an appropriate basis in Qd we may assume that +σ is contained in Qd +⩾0. Moreover, to any σ-polyhedra Q1 and Q2 we may apply a parallel +translation and assume that Q1 and Q2 are contained in the open octant Qd +>0. +Proposition 3. Let Q1 and Q2 be two σ-polyhedra in Qd of full dimension. Then there +are positive integers n and m with d = n − m and a surjective homomorphism π: Zn → Zm +such that Q1 = P(u1) and Q2 = P(u2) for some points u1, u2 ∈ C ∩ Zm lying in a common +GIT-cone. + +NORMALLY LOCATED POLYHEDRA +7 +Proof. Let N (Q1) and N (Q2) be the normal fans of the polyhedra Q1 and Q2, respectively. +Denote by N the coarsest common refinement of the fans N (Q1) and N (Q2). +Let m be the number of rays of the fan N and l1, . . . , lm be linear functions on Qd +generating the rays of N . +Denote by ai (resp. +by bi) the maximal value of li on Q1 +(resp. on Q2). Then the polyhedron Q1 (resp. Q2) is given by inequalities li ⩽ ai (resp. +li ⩽ bi) with i = 1, . . . , m. The fan N is the normal fan of the polyhedron Q1 + Q2, so the +polyhedron Q1 + Q2 is given by inequalities li ⩽ ai + bi, i = 1, . . . , m. +We may assume that the linear functions l1, . . . , lm have integer coefficients. Let n = d+m +and consider the projection π: Zn → Zm given by +(x1, . . . , xd, y1, . . . , ym) → (y1 + l1, . . . , ym + lm). +Let u1 = (a1, . . . , am). Then the polyhedron P(u1) is given by conditions +x1 ⩾ 0, . . . , xd ⩾ 0, y1 ⩾ 0, . . . , ym ⩾ 0, y1 + l1 = a1, . . . , ym + lm = am. +This system is equivalent to +x1 ⩾ 0, . . . , xd ⩾ 0, l1 ⩽ a1, . . . , lm ⩽ am. +It proves that P(u1) = Q1. The same arguments show that +P(u2) = Q2 +and +P(u1 + u2) = Q1 + Q2. +Finally, by Proposition 2 the condition P(u1) + P(u2) = P(u1 + u2) implies that u1 and u2 +lie in a common GIT-cone. +□ +Let us denote by N1 (resp. N2) the normal fan of the polyhedron P(u1) (resp. P(u2)) +living in the space Qd. +Proposition 4. The fan N1 refines the fan N2 if and only if u1 is an interior point of a +GIT-cone containing u2. +Proof. For any point c = (c1, . . . , cn) ∈ Qn, we define a subset in {1, . . . , n} as +Z(c) = {i | ci = 0}. +The condition that u1 is an interior point of a GIT-cone containing u2 means that every +orbit cone, which contains u1, contains u2 as well. Note that the coordinates of a point in +P(u1) (resp. P(u2)) may be considered as coefficients of a linear combination of the vectors +w1, . . . , wn that is equal to u1 (resp. u2). Taking this into account, we may reformulate +the above condition as: for every point v1 ∈ P(u1) there is a point v2 ∈ P(u2) such that +Z(v1) ⊆ Z(v2). +Since we assume that the polyhedra are contained in the open octant Qd +>0, the coordinates +x1, . . . , xd are positive on P(u1) and P(u2). At the same time, for y1, . . . , ym the coordinate +yi is zero at a point v1 (resp. v2) if and only if the linear function li reaches its maximum +ai on P(u1) at v1 (resp. bi on P(u2) at v2). The condition Z(v1) ⊆ Z(v2) means that every +function li which is contained in the cone τ(v1) of the normal fan N1 (see Section 3) is also +contained in the cone τ(v2) of the normal fan N2. Since the rays of the fans N1 and N2 +are generated by some of the vectors li and every cone is uniquely determined by the set of +rays it contains, the last condition means that every cone in N1 is contained in a cone of +N2 or, equivalently, the fan N1 refines the fan N2. +□ +The next theorem generalizes [6, Theorem 1.1] from dimension 2 to higher dimensions. + +8 +IVAN ARZHANTSEV +Theorem 3. Let Q1 and Q2 be two σ-polyhedra in Qd of full dimension. Assume that the +normal fan N (Q1) refines the normal fan N (Q2). Then there exists a positive integer k +such that for every positive integer s the polyhedra skQ1 and skQ2 are normally located. +Proof. Following Proposition 3, we realize Q1 (resp. +Q2) as P(u1) (resp. +P(u2)). +By +Theorem 2, it suffices to show that the point u1 lies in the relative interior of a common +GIT-cone of the points u1 and u2. It follows from Proposition 4. +□ +Remark 3. The construction of two triangles presented in the proof of Proposition 1 is +extracted from Example 1: in notation of this example, we take the triangles P = P(u1) +and Q = P(u2) with respect to the projection π: Z4 → Z2. +References +[1] Ivan Arzhantsev and J¨urgen Hausen. On the multiplication map of a multugraded algebra. Math. Res. +Lett. 14 (2007), no. 1, 129-136 +[2] Florian Berchtold and J¨urgen Hausen. GIT-equivalence beyond the ample cone. Michigan Math. J. 54 +(2006) no. 3, 483-516 +[3] Winfried Bruns, Joseph Gubeladze, and Ngˆo Viˆet Trung. Normal polytopes, triangulations, and Koszul +algebras. J. Reine Angew. Math. 485 (1997), 123-160 +[4] G¨unter Ewald and Uwe Wessels. On the ampleness of invertible sheaves in complete projective toric +varieties. Results Math. 19 (1991), 275-278 +[5] Joseph Gubeladze. Normal polytopes: Between discrete, continuous, and random. J. Pure Appl. Al- +gebra 227 (2023), 107187 +[6] Christian Haase, Benjamin Nill, Andreas Paffenholz, and Francisco Santos. Lattice points in Minkowski +sums. Electron. J. Combin. 15 (2008), N11 +[7] Mariusz Koras and Peter Russell. Linearization problems. In: Algebraic Group Actions and Quotients, +pp. 91-107. Hindawi, Cairo, 2004 +[8] Tadao Oda. Problems on Minkowski sums of convex lattice polytopes. Abstract submitted at the +Oberwolfach Conference “Combinatorial Convexity and Algebraic Geometry” 26.10–01.11, 1997; see +also arXiv:0812.1418, 7 pages +Faculty of Computer Science, HSE University, Pokrovsky Boulevard 11, Moscow, +109028 Russia +Email address: arjantsev@hse.ru + diff --git a/ONE0T4oBgHgl3EQf0gLr/content/tmp_files/load_file.txt b/ONE0T4oBgHgl3EQf0gLr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..07849620a54553c9d92d366940788d4e3e4a334a --- /dev/null +++ b/ONE0T4oBgHgl3EQf0gLr/content/tmp_files/load_file.txt @@ -0,0 +1,375 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf,len=374 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='02688v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='CO] 6 Jan 2023 NORMALLY LOCATED POLYHEDRA IVAN ARZHANTSEV Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Lattice polyhedra Q1 and Q2 with the same tail cone are said to be normally located if every lattice point in the Minkowski sum Q1 + Q2 is the sum of lattice points from Q1 and Q2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' We prove that if the normal fan of Q1 refines the normal fan of Q2, then there is a positive integer k such that for any positive integer s the polyhedra skQ1 and skQ2 are normally located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' This result is based on an interpretation of the problem in terms of graded algebras and earlier results on surjectivity of the multiplicaiton map on homogeneous components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Also we provide an example of two lattice triangles P and Q on the plane such that for any positive integer k the triangles kP and kQ are not normally located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Introduction Let us consider the lattice Zd and the rational vector space Qd generated by Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' By a lattice polytope we mean a convex polytope P in Qd with vertices in Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let us assume additionally that the lattice points in P generate the lattice Zd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' this can be achieved by replacing Zd with a proper sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' It is easy to construct examples of two lattice polytopes P and Q such that the Minkowski sum P + Q contains a lattice point that is not a sum of lattice points from P and Q, respective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=', [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Moreover, starting from dimension 3 it may happen even when P = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' A lattice polytope P is called normal if for every positive integer s and every lattice point z ∈ sP there are lattice points z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , zs ∈ P such that z = z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' + zs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Normal polytopes play an important role in many areas of modern mathematics, see [5] for a recent survey on this subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' In particular, such polytopes define integrally closed graded monoid algebras and projectively normal embeddings of projective toric varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let us recall that a lattice polytope P is smooth if the primitive edge vectors at every vertex of P form a basis of Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Smooth polytopes correspond to projective embeddings of smooth toric varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Oda’s question [5, Section 1] asks whether every smooth polytope is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' This question is open in all dimensions ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The proof of the following theorem may be found in [3, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='3] or [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' For every lattice polytope P in Qd the polytope (d − 1)P is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' In particular, in dimension 2 any lattice polygon is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The aim of this paper is to generalize the property of normality to a pair of polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Lattice polytopes P and Q in Qd are said to be normally located if for every lattice point z ∈ P + Q there are lattice points z′ ∈ P and z′′ ∈ Q such that z = z′ + z′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Primary 11P21, 52B20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Secondary 14M25, 52B11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Polytope, polyhedron, lattice, Minkowski sum, graded algebra, toric variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' 1 2 IVAN ARZHANTSEV Oda’s Conjecture (see [8] or [6, Section 1]) states the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let P and Q be lattice polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Assume that P is smooth and the normal fan of Q refines that of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Then the polytopes P and Q are normally located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' This is a generalization of Oda’s question on normality of a smooth polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' In [6, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='1], it is shown that if P and Q are lattice polygons such that the normal fan of Q refines that of P, then P and Q are normally located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' In this paper we prove a version of Oda’s Conjecture (Theorem 3) which generalizes [6, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='1] to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The result is obtained not only for polytopes, but also for polyhedra with the same tail cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' In Section 2 we give an explicit example of two lattice triangles P and Q on the plane such that for any positive integer k the triangles kP and kQ are not normally located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' This example is intended to demonstrate that, unlike the normality property, the property of normal location cannot always be achieved just by rescaling two given polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' In Section 3 we recall basic definitions and facts on polyhedra, their Minkowski sums and normal fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Also we consider polyhedra that appear as fibers of a projection of the positive octant in a bigger lattice to a smaller lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Section 4 is devoted to interpretations of the objects defined above in terms of graded algebras and the multiplication map on homogeneous components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' We recall the results of [1] that allow to relate the property of normal location (up to scalar) of a pair of polyhedra in fibers over two points with the location of these points with respect to the so-called GIT-fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Finally, in Section 5 we explain how to realize a pair of polyhedra via the fiber construction and prove our main result (Theorem 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' An example in dimension 2 Let us show that, in contrast to Theorem 1, for two lattice triangles absence of the property of normal location is not just a question of scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' There are two lattice triangles P and Q in Q2 such that for every positive integer k the triangles kP and kQ are not normally located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Take P = Conv(a1, a2, a3) and Q = Conv(b1, b2, b3), where a1 = (165, 0), a2 = (175, 0), a3 = (0, 385) and b1 = (0, 0), b2 = (35, 0), b3 = (0, 77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Equivalently, the triangle P is given by inequalities 7x + 3y ⩾ 1155, 11x + 5y ⩽ 1925, y ⩾ 0 and Q is given by 11x + 5y ⩽ 385, x ⩾ 0, y ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The Minkowski sum P + Q is Conv(c1, c2, c3, c4), where c1 = (165, 0), c2 = (210, 0), c3 = (0, 385), c4 = (0, 462).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The inequalites that determine P + Q are 7x + 3y ⩾ 1155, 11x + 5y ⩽ 2310, x ⩾ 0, y ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' It is easy to check that for every positive integer k the point s = (1, 385k − 2) is contained in kP + kQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Assume that s = p + q, where p ∈ kP ∩ Z2 and q ∈ kQ ∩ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Then only two cases are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' NORMALLY LOCATED POLYHEDRA 3 Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let p = (1, a) and q = (0, 385k − 2 − a) for some non-negative integer a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Since p is contained in P, we have 7 + 3a ⩾ 1155k and 11 + 5a ⩽ 1925k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' These inequalities may be rewritten as a ⩾ 385k − 2 − 1 3 and a ⩽ 385k − 2 − 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Since a is integer, we come to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let p = (0, a) and q = (1, 385k − 2 − a) for some non-negative integer a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Since p is contained in P, we have 3a ⩾ 1155k and 5a ⩽ 1925k, so a ⩾ 385k and a ⩽ 385k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' We conclude that a = 385k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Since q lies in Q, we have 385k − 2 − a = −2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' These two contradictions complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Generalities on polyhedra By a polyhedron in Qd we mean the intersection of finitely many closed affine half spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' A lattice polyhedron is a polyhedron in Qd whose vertices are in Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Note that a polytope can be defined as a bounded polyhedron in Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' For a polyhedron Q we define its relative interior as the set obtained by removing all proper faces from Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let us denote the relative interior of Q by Q◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Further, a cone in Qd is the intersection of finitely many closed linear half spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' A cone is pointed if it contains no line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' If a cone has dimension at least 2, it is pointed if and only if it is generated by its one-dimensional faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The set of all polyhedra in Qd comes with a natural structure of a commutative semigroup: one defines the Minkowski sum of two polyhedra Q1 and Q2 to be the polyhedron Q1 + Q2 := {w1 + w2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' w1 ∈ Q1, w2 ∈ Q2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' In the same way one may define the Minkowski sum of two arbitrary subsets in Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Any polyhedron allows a Minkowski sum decomposition Q = P +σ, where P is a polytope and σ is a cone in Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' In this decomposition, the tail cone σ is unique;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' it is given by σ = {w ∈ Qd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' w′ + tw ∈ Q for all w′ ∈ Q, t ∈ Q⩾0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' A polyhedron with the tail cone σ is called a σ-polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' For example, polytopes are precisely σ-polyhedra with σ = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' It is easy to check that for a fixed cone σ the set of all σ-polyhedra forms a commutative semigroup with respect to the Minkowski sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let us recall that with any σ-polyhedron Q in Qd one may associate the normal fan N (Q): any point v ∈ Q defines the cone τ(v) consisting of all linear functions on Qd which reach their maximal value on Q at the point v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The collection of cones N (Q) = {τ(v) | v ∈ Q} is finite and it is a fan in a sense that a face of any cone in N (Q) is contained in N (Q) and the intersection of any two cones in N (Q) is a face of each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Moreover, all cones in N (Q) are pointed if and only if Q has full dimension in Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' 4 IVAN ARZHANTSEV The support of a fan N is the union of all cones in N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The support of the normal fan N (Q) equals the dual cone σ∨ := {l ∈ (Qd)∗ | l|σ ⩾ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' We say that a fan N1 refines a fan N2, if every cone in N1 is contained in some cone in N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let N1 and N2 be two fans with the same support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The coarsest common refinement of N1 and N2 is the fan N with the same support, whose cones are τ1 ∩ τ2, where τ1 ∈ N1 and τ2 ∈ N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' It is well-known that for normal fans N (Q1) and N (Q2) of two σ-polyhedra Q1 and Q2 the coarsest common refinement N coincides with the normal fan N (Q1 + Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Now let us consider a surjective homomorphism of lattices π: Zn → Zm and the induced linear map of vector spaces π: Qn → Qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' We denote by Qn ⩾0 the cone of vectors in Qn with non-negative coordinates and let C := π(Qn ⩾0) ⊆ Qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Consider the cone σ := π−1(0)∩Qn ⩾0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' With any point u ∈ C one associates the polyhedron P(u) := π−1(u) ∩ Qn ⩾0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' For any u ∈ C the polyhedron P(u) is a σ-polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' For a vector w ∈ Qn, the condition w′ + tw ∈ P(u) for all w′ ∈ P(u) and all t ∈ Q⩾0 means that w ∈ Qn ⩾0 and u = π(w′ + tw) = π(w′) + π(tw) = u + tπ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' It is equivalent to w ∈ Qn ⩾0 and π(w) = 0, or w ∈ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' □ Clearly, for every u ∈ C there is a positive integer r such that P(ru) is a lattice poly- hedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Also it is easy to check that P(u1) + P(u2) is contained in P(u1 + u2) for all u1, u2 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' We are interested in the following three properties of a pair (u1, u2) with u1, u2 ∈ C: (P1) P(u1) + P(u2) = P(u1 + u2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' (P2) (P(u1) ∩ Zn) + (P(u2) ∩ Zn) = P(u1 + u2) ∩ Zn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' (P3) there exists k ∈ Z>0 such that for any s ∈ Z>0 we have (P(sku1) ∩ Zn) + (P(sku2) ∩ Zn) = P(sk(u1 + u2)) ∩ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Graded algebras In this section we introduce an algebraic interpretation of the objects discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The projection π: Zn → Zm gives rise to an effective Zm-grading on the polynomial algebra A := K[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , xn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Namely, we put deg(xi) = π(ei), where e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , en is the standard basis of the lattice Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' For further purposes we assume the ground field K to be an algebraically closed field of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The Linearization Problem [7] claims that up to automorphism any effective Zm-grading on K[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , xn] is obtained this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Below we follow the presentation given in [1] in a somewhat more general situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let A be an associative, commutative, integral, finitely generated algebra with unit over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Suppose that A is graded by the lattice Zm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=', we have A = � u∈Zm Au with Au1 · Au2 ⊆ Au1+u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' NORMALLY LOCATED POLYHEDRA 5 By the weight cone of A we mean the cone C(A) ⊆ Qm generated by all u ∈ Zm with Au ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' We investigate the following problem: given u1, u2 ∈ C(A) ∩ Zm, does there exist an integer k > 0 such that for any s > 0 the multiplication map Asku1 ⊗K Asku2 → Ask(u1+u2), f ⊗ g �→ fg is surjective?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' We call a pair u1, u2 ∈ C(A) ∩ Zm generating if it has this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' If A is a polynomial algebra with Zm-grading given by the projection π, this is precisely property (P3) for a pair (u1, u2) of lattice points in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let us recall from [2] the concept of the GIT-fan associated to a graded algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' A Zm- grading on A defines an action of the torus T := Spec(K[Zm]) on X := Spec(A) such that for any u ∈ Zm, the elements f ∈ Au are precisely the semiinvariants of the character χu : T → K∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=', each f ∈ Au satisfies f(t·x) := χu(t)f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The orbit cone of a (closed) point x ∈ X is the cone ω(x) ⊆ Qm generated by all u ∈ C(A) admitting an f ∈ Au with f(x) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The collection of orbit cones is finite, and thus one may associate to any element u ∈ C(A) its GIT-cone: λ(u) := � x∈X, u∈ω(x) ω(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' These GIT-cones cover the weight cone C(A) and, by [2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='11], the collection Λ(A) of all GIT-cones is a fan in the sense that if λ ∈ Λ(A) then also every face of λ belongs to Λ(A), and for τ, λ ∈ Λ(A) the intersection τ ∩ λ is a face of both λ and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Note that we allow here a fan to have cones containing lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' [1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='1] (1) If u1, u2 ∈ C(A) ∩ Zm is a generating pair, then the weights u1, u2 lie in a common GIT-cone λ ∈ Λ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' (2) If u1, u2 ∈ C(A) ∩ Zm lie in a common GIT-cone λ ∈ Λ(A) and u1 belongs to the relative interior λ◦, then (u1, u2) is a generating pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' If two weights u1, u2 ∈ C(A) ∩ Zm lie on the boundary of a common GIT-cone λ ∈ Λ(A), then no general statement in terms of the GIT-fan is possible: it may happen that u1, u2 is generating, and also it may happen that u1, u2 is not generating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' For the first case there are obvious examples, and for the latter we present the following one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' [1, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='2] Consider the polynomial ring A := K[x1, x2, x3, x4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Then one may define a Z2-grading on A by setting deg(x1) := (4, 1), deg(x2) := (2, 1), deg(x3) := (1, 2), deg(x4) := (1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The pair u1 := (2, 1) and u2 := (1, 2) is contained in a common GIT-cone but it is not generating: one checks directly that the monomials x1xs−2 2 xs−1 3 x4 ∈ As(u1+u2) can never be obtained by multiplying elements from Asu1 and Asu2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' In [1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='5], a criterion for a pair of weights (u1, u2) in one GIT-cone to be generating is given in terms of normality of the image of a morphism between certain quotient spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The proof of [1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='1] is based on several propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' A modification of one of them will be used below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' To formulate this modification, we need some more notions from the theory of graded algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' 6 IVAN ARZHANTSEV A subalgebra B of a graded algebra A = ⊕u∈ZmAu is homogeneous if B is the direct sum of intersections of B with homogeneous components of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Every homogeneous subalgebra in A inherits a Zm-grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' With any weight u ∈ Zm one associates a homogeneous subalgebra A(u) := ⊕r⩾0Aru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Note that the subalgebra A(u) is Z⩾0-graded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Further, with any weights u1 and u2 one associates a homogeneous subalgebra A(u1, u2) in A(u1 + u2) defined as A(u1, u2) := � r⩾0 Aru1 · Aru2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' We say that a homogeneous subalgebra B of a Z⩾0-graded algebra A is big, if the radical of the ideal B+ := ⊕r>0Br coincides with A+ := ⊕r>0Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let A be the polynomial algebra K[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , xn] with Zm-grading given by a projection π: Zn → Zm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' For every v ∈ Zn ⩾0 we denote by xv the monomial xv1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' xvn n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Condition (P1) on a pair of weights (u1, u2) is equivalent to each of the conditions: (1) the weights u1, u2 lie in a common GIT-cone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' (2) the subalgebra A(u1, u2) is big in A(u1 + u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The equivalence of conditions (1) and (2) is proved in [1, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let us prove that (P1) implies (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Assume that P(u1 + u2) = P(u1) + P(u2) and take a monomial xv ∈ A(u1 +u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' We have to prove that xv is contained in the radical of the ideal A(u1, u2)+, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=', there is a positive integer q such that xqv ∈ A(u1, u2)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' By assumption, we have v = v′ + v′′ with v′ ∈ P(u1) and v′′ ∈ P(u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Then there is q ∈ Z>0 such that qv = qv′ + qv′′ with qv′ ∈ P(u1) ∩ Zn and qv′′ ∈ P(u2) ∩ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' This proves the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Now we come to implication (2) ⇒ (P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Note that v ∈ P(u1 + u2) if and only if there is s ∈ Z>0 such that sv ∈ P(s(u1 + u2)) ∩ Zn or, equivalently, the monomial xsv lies in A(u1 + u2)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' By assumption, this implies that xsv is contained in the radical of A(u1, u2)+, or there exists t ∈ Z>0 such that xtsv ∈ A(u1, u2)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The later condition means that there are v′ ∈ P(tsu1)∩Zn and v′′ ∈ P(tsu2)∩Zn such that tsv = v′ +v′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' This condition implies v ∈ P(u1) + P(u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Positive results on normal location We keep the notation introduced in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' In particular, every surjec- tive homomorphism of lattices π: Zn → Zm gives rise to a Zm-grading on the algebra K[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , xn], and we speak about the weight cone, the orbit cones and the GIT-cones cor- responding to this grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' In this situation, the weight cone coincides with C = π(Qn ⩾0) and the orbits cones are precisely the cones generated by all subsets of the set {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , wn}, where wi := π(ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let σ be a pointed cone in Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Taking an appropriate basis in Qd we may assume that σ is contained in Qd ⩾0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Moreover, to any σ-polyhedra Q1 and Q2 we may apply a parallel translation and assume that Q1 and Q2 are contained in the open octant Qd >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let Q1 and Q2 be two σ-polyhedra in Qd of full dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Then there are positive integers n and m with d = n − m and a surjective homomorphism π: Zn → Zm such that Q1 = P(u1) and Q2 = P(u2) for some points u1, u2 ∈ C ∩ Zm lying in a common GIT-cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' NORMALLY LOCATED POLYHEDRA 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let N (Q1) and N (Q2) be the normal fans of the polyhedra Q1 and Q2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Denote by N the coarsest common refinement of the fans N (Q1) and N (Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let m be the number of rays of the fan N and l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , lm be linear functions on Qd generating the rays of N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Denote by ai (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' by bi) the maximal value of li on Q1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' on Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Then the polyhedron Q1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Q2) is given by inequalities li ⩽ ai (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' li ⩽ bi) with i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The fan N is the normal fan of the polyhedron Q1 + Q2, so the polyhedron Q1 + Q2 is given by inequalities li ⩽ ai + bi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' We may assume that the linear functions l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , lm have integer coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let n = d+m and consider the projection π: Zn → Zm given by (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , xd, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , ym) → (y1 + l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , ym + lm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let u1 = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , am).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Then the polyhedron P(u1) is given by conditions x1 ⩾ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , xd ⩾ 0, y1 ⩾ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , ym ⩾ 0, y1 + l1 = a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , ym + lm = am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' This system is equivalent to x1 ⩾ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , xd ⩾ 0, l1 ⩽ a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , lm ⩽ am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' It proves that P(u1) = Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The same arguments show that P(u2) = Q2 and P(u1 + u2) = Q1 + Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Finally, by Proposition 2 the condition P(u1) + P(u2) = P(u1 + u2) implies that u1 and u2 lie in a common GIT-cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' □ Let us denote by N1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' N2) the normal fan of the polyhedron P(u1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' P(u2)) living in the space Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The fan N1 refines the fan N2 if and only if u1 is an interior point of a GIT-cone containing u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' For any point c = (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , cn) ∈ Qn, we define a subset in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , n} as Z(c) = {i | ci = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The condition that u1 is an interior point of a GIT-cone containing u2 means that every orbit cone, which contains u1, contains u2 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Note that the coordinates of a point in P(u1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' P(u2)) may be considered as coefficients of a linear combination of the vectors w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , wn that is equal to u1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Taking this into account, we may reformulate the above condition as: for every point v1 ∈ P(u1) there is a point v2 ∈ P(u2) such that Z(v1) ⊆ Z(v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Since we assume that the polyhedra are contained in the open octant Qd >0, the coordinates x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , xd are positive on P(u1) and P(u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' At the same time, for y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' , ym the coordinate yi is zero at a point v1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' v2) if and only if the linear function li reaches its maximum ai on P(u1) at v1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' bi on P(u2) at v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The condition Z(v1) ⊆ Z(v2) means that every function li which is contained in the cone τ(v1) of the normal fan N1 (see Section 3) is also contained in the cone τ(v2) of the normal fan N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Since the rays of the fans N1 and N2 are generated by some of the vectors li and every cone is uniquely determined by the set of rays it contains, the last condition means that every cone in N1 is contained in a cone of N2 or, equivalently, the fan N1 refines the fan N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' □ The next theorem generalizes [6, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='1] from dimension 2 to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' 8 IVAN ARZHANTSEV Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Let Q1 and Q2 be two σ-polyhedra in Qd of full dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Assume that the normal fan N (Q1) refines the normal fan N (Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Then there exists a positive integer k such that for every positive integer s the polyhedra skQ1 and skQ2 are normally located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Following Proposition 3, we realize Q1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Q2) as P(u1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' P(u2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' By Theorem 2, it suffices to show that the point u1 lies in the relative interior of a common GIT-cone of the points u1 and u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' It follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' The construction of two triangles presented in the proof of Proposition 1 is extracted from Example 1: in notation of this example, we take the triangles P = P(u1) and Q = P(u2) with respect to the projection π: Z4 → Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' References [1] Ivan Arzhantsev and J¨urgen Hausen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' On the multiplication map of a multugraded algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' 14 (2007), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' 1, 129-136 [2] Florian Berchtold and J¨urgen Hausen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' GIT-equivalence beyond the ample cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Michigan Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' 54 (2006) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' 3, 483-516 [3] Winfried Bruns, Joseph Gubeladze, and Ngˆo Viˆet Trung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Normal polytopes, triangulations, and Koszul algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' 485 (1997), 123-160 [4] G¨unter Ewald and Uwe Wessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' On the ampleness of invertible sheaves in complete projective toric varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Results Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' 19 (1991), 275-278 [5] Joseph Gubeladze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Normal polytopes: Between discrete, continuous, and random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Al- gebra 227 (2023), 107187 [6] Christian Haase, Benjamin Nill, Andreas Paffenholz, and Francisco Santos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Lattice points in Minkowski sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' 15 (2008), N11 [7] Mariusz Koras and Peter Russell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Linearization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' In: Algebraic Group Actions and Quotients, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' 91-107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Hindawi, Cairo, 2004 [8] Tadao Oda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Problems on Minkowski sums of convex lattice polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' Abstract submitted at the Oberwolfach Conference “Combinatorial Convexity and Algebraic Geometry” 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='10–01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='11, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content=' see also arXiv:0812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='1418, 7 pages Faculty of Computer Science, HSE University, Pokrovsky Boulevard 11, Moscow, 109028 Russia Email address: arjantsev@hse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} +page_content='ru' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE0T4oBgHgl3EQf0gLr/content/2301.02688v1.pdf'} diff --git a/OtE0T4oBgHgl3EQfjwFX/content/tmp_files/2301.02463v1.pdf.txt b/OtE0T4oBgHgl3EQfjwFX/content/tmp_files/2301.02463v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4da990d413748ddf415964471b2089d1b0a7a54b --- /dev/null +++ b/OtE0T4oBgHgl3EQfjwFX/content/tmp_files/2301.02463v1.pdf.txt @@ -0,0 +1,1648 @@ + + The exact description of intruder states in 112Cd nucleus by using a mixing +formalism based on SU(1,1( transitional Hamiltonian and O(6)–Casimir +operator + + +M. Rastegar*, H. Sabri ,A. O. Ezzati + +Department of Physics, University of Tabriz, 51664 Tabriz, Iran + + + + + + + + + + + + + + + + + + + + + + + + + + +* Corresponding author e-mail: m.rastgar@tabrizu.ac.ir + + +2 + + +Abstract; +In this paper, we used a transitional Hamiltonian which has U(5)↔O(6) transition to improve +theoretical predictions for energy spectra and quadrupole transition rates of 112Cd nucleus. To +this aim, the transitional Hamiltonian in the affine SU(1,1) algebra has been extended by adding +the O(6)-Casimir operator and mixing Hamiltonian to increase exactness in the description of +24 + and +32 + intruder levels of this nucleus. We also considered the wave functions of both +regular and intruder states, as a combination in the N and N+2 boson spaces. The results +confirm the advantages of using such mixing approaches and describing the energy and +transition rates with high accuracy. +Keywords: Interacting boson model (IBM), particle-hole excitation, intruder levels, transitional +Hamiltonian, quadrupole transition rates. + +1. Introduction +One of the genius nuclei that are in the transitional between vibrational and axial symmetry +regions are 110,112,114Cd but in this paper, we want to test just 112Cd nucleus because it seems +great exhibiting for our method and also Lehman in the Ref.[19].emphasize it. Also in Ref [35] +mentioned about the particle-hole excitations in 108Cd by the intruder states that just worked by +Hamiltonian. There is a novelty way in this paper that it is work by expansion wave function. +The intruder levels and their observation via experimental techniques, with descriptions of +these levels by the theoretical framework, are the subject of different studies. The work of Bohr +and Kalkar in nuclear physics has exposed the intention of deformed shapes and the +manifestation of disparate shapes in a given nucleus [1]. This idea has been developed by other +original studies, such as [2-5], which used different shapes in a nucleus for a complete +description of its observables. Various models based on the shell model's concepts, collective +geometric model, models based on the mean-field approach with algebraic frameworks are +used for such investigations. On the other hand, the multi particles multi holes (mp-mh) is +another exclusive method which first proposed in the shell model framework and then +expanded in the algebraic models such as the interacting boson model (IBM) to describe +different intruder levels in nuclei that are located near the closed proton and neutron shells. The +idea of expanding the ordinary 2p-2h intruder excitations into two pairs of extra nucleons (two + + +3 + +bosons) and studying their mixing with the regular configurations that first Dual and Barrett +presented this approach [2]. It has been widely reported that intruder states have been exposed +to low excitation energies [4-10,11,12]. mp-mh excitations cannot be compounded easily in +complete large-scale shell-model studies because of the extensive extents of the model spaces +involved. The new label has been introduced to represent the symmetries connecting particle +and hole bosons within the IBM configuration mixing calculations [13-17]. In this approach, +the effect of such excitations is expressed; as different terms of regular Hamiltonian and, +therefore; a combination of regular and mixing Hamiltonians e.g.; other symmetries, improve +theoretical predictions for both regular and intruder states. A transitional Hamiltonian between +different symmetry limits of IBM provides a similar combination of different symmetries, and +this idea is the main subject of this investigation. +A detailed investigation of both regular and intruder levels of the Cd isotopic chain has been +done by Lehman et al. in Ref.[18]. They have claimed that the vibrational characteristics of +112Cd portray by the regular configuration subtending of N bosons, s, and d bosons. As a result +of the more deformation nature of the intruder configuration, there are two extra bosons +involved, N+2, which takes account of a 2p-2h excitation within the closed shells [18-19]. +They showed that the +2 +4  and +3 +2  are intruder states for U(5) algebra, and as anticipated, this limit +is too narrow to duplicate the whole set of empirical data, as some low-lying states are outside +the model space [20]. To improve the contract between theoretical and experimental results, a +subsequent configuration called the intruder configuration had been required. Therefore, they +have combined the regular and intruder configurations, which correspond with U(5) and O(6) +dynamical limits of IBM, respectively, for N and N+2 bosons spaces. +In this investigation, we extend a similar combination by using a transitional Hamiltonian +between U(5) and SO(6) limits which are based on affine SU(1,1) algebra [16-17] as a regular +Hamiltonian in the N boson space. In recent studies, this Hamiltonian was used to explain the +coexistence of deformed shapes by using the two control parameters of transitional +Hamiltonian. Then, this Hamiltonian mixed with the O(6) Casimir operator in the N+2 bosons +space to describe intruder levels of 112Cd nucleus with high accuracy. We also considered the +advantages of this mixed formalism in describing of different quadrupole transition rates of +this nucleus. +2. Method; + + +4 + +2.1. The regular part of Hamiltonian (unperturbed representation): +For the nuclei in the vicinity of the proton closed shell at Z = 50, such as Cd isotopic chain, the +regular configuration can be exposed by U(5) algebra in the IBM model, and the intruder +configuration gives away the O(6) symmetry [18]. Intruding, the construction owning to the +particle-hole excitation amidst the closed shells is a well-known particularity in the nucleus. +This configuration is a new implement for the exposition of the level scheme of nuclear that +has not been predicted in one algebra. In; cases where the Hamiltonian could be written with +the help of the limits of the Casten triangle, the Hamiltonian could be written with invariants +operators of two of the three chains [24,25]. One of these transitions is U(5)↔O(6) transition +which is used to describe nuclei between vibrational and gamma-unstable limits, nuclei near +the closed shells. In the framework of IBM, the even-even Cd isotopes are good candidates to +perform the such a transition. This means one may use such transitional Hamiltonian to +investigate energy spectra and transition rates of nuclei in this region. The advantages of +SU(1,1)-based transitional Hamiltonian in comparison with other algebraic approaches which +are illustrated in detail in Refs.[17-18]. Encourage us to focus only on this method. We used +this SU(1,1) transitional Hamiltonian by the definition of [N] boson presentation; details, about +this transitional Hamiltonian, are available in Refs. [17-18,20]. Here, we briefly introduce the +main concepts and relations. +The SU(1,1) algebra has been constructed by +, +0, +S   + operators in which the generators +d  boson pairing algebra are: +† +† +0 +† +† +1 +1 +1 +( ) +( +. +) , +( ) +( . ) , +( ) +( +) (1) +2 +2 +4 +S +d +d +d +S +d +d d +S +d +d d +d d + + + + + + + + + + + + +And these operators for s  boson pairing algebra define as: +†2 +2 +0 +† +† +1 +1 +1 +( ) + , +( ) + , +( ) +( +) (2) +2 +2 +4 +S +s +s +S +s +s +S +s +s s +ss + + + + + + +The infinite-dimensional SU(1,1) algebra has been generated by using [17,18]: +2 +1 +2 +1 +0 +2 +0 +2 +0 +( ) +( ) , +( ) +( ) (3) +n +n +n +n +n +s +d +n +s +d +S +c +S +s +c +S +d +S +c S +s +c S +d + + + + + + + + + + +Where cs and cd are real parameters and n can be 0, 1, 2,... +  +. The commutation relations +between these operators are: + + +5 + +0 +0 +1 +[ +, +] + , [ +, +] +2 + (4) +m +n +m n +m +n +m n +S +S +S +S +S +S + + + + + +  +  +  +Then, affine Lie algebra SU(1,1) without central extension has been constructed by using these +generators. The; Casimir operator is +0 +0 +2ˆ +( +1) +C +S +S +S  + + + +, too. +The IBM-1 Hamiltonian, without distinguishing between proton and neutrons, in this +formalism, has written as: +0 +(1,1) +0 +0 +1 +2ˆ +ˆ +ˆ +ε +[( +(5) +(3)] . +regular +SU +H +H +gS S +S +C +SO +SO + + + + + + + + + + ( ε , γ , δ and g are real +parameters) (5) +Cs = Cd corresponds with the SO(6) limit. Also, the U(5) limit has been described by the cs = 0 +requirements, and finally, cs ≠ cd has been used for the explanation of the U(5)↔O(6) +transitional region. We present the regular or unperturbed states of SU(1,1) algebra by +with +ix number parameters c  n terms of unknown , dwhich define +, ; +s +N k +n LM +  + + as:, +1,2,..., +i +k + + +1 +2 +1 +2 +1 +2 +1 +2 +, ; +... +... +... + , (6) +k +k +k +i +n +n +n +s +n +n +n +k +n +n +n +n +Z +N k +n LM +a a +a x +x +x +S S +S +lw +  + + + + + +  + +k relate to the total number of boson N as N = 2k + ν + νs. The analytical behavior of wave +functions allows us to consider +ix near zero. Now, the wave functions of SU(1,1) algebra are +yielded by using the commutation relations between the generators as: +1 +2 +, ; +... + , (7) +k +s +x +x +x +N k +n LM +NS S +S +lw +  + + + + + +Where N is the normalization factor and +2 +2 +( ) +( ) , (8) +1 +1 +i +s +d +x +s +i +d +i +c +c +S +S +s +S +d +c x +c x + + + + + + + +Also, we defined the xi values via the following set of equations: +2 +2 +2 +2 +1 +5 +( +) +( +) +ε +2 +2 +2 + +- + , + +1,2,..., + (9) +1 +1 +s +s +d +i +j +i +s +i +d +i +i +j +gc +gc +for i +k +x +c x +c x +x +x + + + + + + + + + + + + +The eigenvalues of Eq.5, regular Hamiltonian by 0p-0h excitation, are determined as: + +  +(1,1) +ˆ +, ; +, ; + , (10) +k +s +SU +s +E +N k +n LM H +N k +n LM +  +  + + + + +The final form is: + + +6 + +  +  + + + + +  +0 +1 +0 +2 +2 +1 +1 +3 +1 +1 +1 +5 + and +2 +2 +2 +k +k +k +k +s +d +i +i +E +h +L L +c +c +h +s +x +  + + + + + + + + + + + +  + + + + + + +  + + + + + + + + + + + + + + + + + + (11) +Now, we must extract the parameters of this equation in comparison with experimental data +for energy levels of 112Cd nucleus. To this aim firstly, we selected, +ε +g +  +, g = 1 keV, +1 +s +d +c +c +c + + + +similar to adn alautc values of ase wit rofhc +1 +i  ) have solved for8then, Eq. ( +2 +yi +d +i +c x + +and +what have done in Refs.[20]. After getting the roots for each level, and m identical to what that +have done in Refs.[20].  and +s + combination into h(k), we can use some extraction process such +as least square fitting are other global methods but to get the exact results we used the roots of +equations by beth-anthats methods, we follow the method that, used the least square fit in +Matlab software to extract δ and γ [21]. These processes, have been repeated by using the +different values of c and ε to reduce the difference between theoretical predictions for different +regular and intruder energy levels and experimental counterparts. We changed the cs values +between 0 and 1 limits with Δcs = 0.1 step length; determined all roots, parameters, and energy +values for each case; some of these results have presented in Table1.We also used the + + +2 +exp +2 +1 +Th +i +E +E +N +  + + +, N, the number of considered levels, as the quality measure for our +extracting procedure, which their values for different cs are present in Table 1, too. +Table1. Regular; and intruder energy levels of 112Cd nucleus have been determined by using different cs values +with related quantum numbers in the framework of SU(1,1) algebra. The experimental data are taken from Ref. +[28]. σ describes the quality of extraction processes. All; of the energy values are in keV. + +Level k ν νs Eexp Eth(cs = 0) Eth(cs = 0.2) Eth(cs = 0.4) Eth(cs = 0.6) Eth(cs = 0.8) Eth(cs = 1) + + +10 4 0 0 0 0 0 0 0 0 0 + +12 3 2 0 617 594 580 600 741 801 875 + +2 +0 3 1 1 1224 1297 1311 1328 1362 1375 1411 + +2 +2 3 1 1 1312 1374 1785 1789 1801 1852 1940 + +14 3 2 0 1415 1576 1800 1852 1889 1885 1912 + +3 +0 2 4 0 1433 1389 1274 1526 1621 1785 1796 + + +7 + + +3 +2 2 4 0 1468 1517 1540 1559 1567 1595 1507 + +4 +0 2 2 2 1870 1890 2088 2211 2321 2425 2611 + +2 +4 3 1 1 1871 1954 1980 1997 2007 2041 1962 + 13 3 2 0 2065 2123 2310 2221 2450 2778 2913 + +3 +4 2 4 0 2082 2169 2105 2249 2271 2301 2381 + +4 +2 2 2 2 2122 2188 2209 2301 2285 2264 2350 + +5 +2 2 1 3 2156 2204 2270 2351 2384 2224 2370 + +16 3 2 0 2168 2214 2454 2574 2679 2321 2376 + + σ 76.16 86.16 92.99 108.11 270.25 321.66 + +Our results show good predictions of the transitional Hamiltonian, independent of cs values. +We examine different values of cs and the best agreement between theoretical and experimental +counterparts found in cs = 0.4.This confirms our idea of using such transitional formalism as a +regular part of this nucleus. These results are in agreement with the predictions of Jolie et al. +[31-34] and Heyde et al. [31-32], which showed the advantages of U(5) dynamical symmetry +in the description of the regular part in the energy spectra of the Cd isotopic chain, respectively +shell model and a combination of U(5) and O(6) dynamical limits. Also, only for +2 +4  and +3 +2  as +intruder levels, the distance between theoretical predictions and experimental values are +apparent, and for other levels in ground and excited bands, this formalism makes appropriate +results. Another; point in this table is the better results of the cs = 0 case, which, as we have +expected for the N bosons space, the U(5) limit excepted for such space and also described +regular states with high accuracy. This result confirms the predictions of Lehman et al in +Ref.[7,19], which used the Hamiltonian of U(5) limit for the regular states of this nucleus. In +the following, we add new terms to our regular Hamiltonian and also suppose the N+2 bosons +space in determining of the effects of these new terms to make an accurate description of +intruder levels. This idea, adding the O(6) limit as a perturbation to the regular U(5) +Hamiltonian, is similar to the method that Heyde et al. [31,32], have proposed because both +groups have a standard O(5) subgroup. +2.2. Additional terms +The affine SU(1,1) lie algebra, has been used here to provide a relevantly simple solvable +pairing model that incorporates the mixing of two-particle and two-hole configurations. This + + +8 + +transitional Hamiltonian in our calculations in the previous subsection works on the states of +N bosons space described as +, ; +s +N k +n LM +  + +. Now, we introduce the method based on the +extension of this transitional Hamiltonian by adding the Casimir operator of O(6) dynamical +limit and also carrying out the calculation in the N+2 space, which the total wave function with +the +2, ; +s +N +k +n LM +  + + + type. In this way, we followed the method developed by Lehman and +Jolie, with the nature of +2 +4  and +3 +2  states, as have been predicted by Heyde et al. [22,31,32]. +Rowe, in Refs. [22,24] has claimed that ˆ ˆ +S S + + the operator in the framework of SU(1,1) algebra +is the Casimir operator of O(6) symmetry. We extended our approach by adding the ˆ ˆ +S S + + term +to affine SU(1,1) algebra as 2p-2h excitation, similar to the method done by Lehman in +Ref.[18]; In the framework of U(6) algebra. Another term that must add to our extended +Hamiltonian is the mixing term which combines both N and N+2 bosons spaces. To this aim, +we used the usual mixing Hamiltonian, which is defined in different papers as: +0 +0 +ˆ +[ +] +[ +] +mix +H +s +s +s +s +d +d +d +d + + + + + + + + +  + + + + + (12) +We also used that our considered transitional Hamiltonian connects both U(5) and O(6) +dynamical limits by the variation of a control parameter of transitional Hamiltonian and, +therefore; has the same role. The Hamiltonian to illustrate mixing could be written as follows: +  + + +(1,1) +2 +1 +2 +(1,1) +2 +ˆ +ˆ ˆ +ˆ +ˆ +ˆ +ˆ +( +) +( +6 +) +( +) +tot +N +SU +N +N +N +N +SU +mix +N +H +P +H +P +P +gC O +gS S +P +P +H +H +P + + + + + + + + + + + (13) +The; PN is the projection operator that represents the N-boson subspaces, while PN+2 projects +the subspaces to N + 2 bosons [18-19]. + To determine the energy levels for these new extended Hamiltonian, we used again Eq.(10), +where the considered +i + + states to get eigenvalues are yielded by combining the two wave +functions as: +2 +' +i +N +N + +  +  + + + +. +The +eigenvalues +of +this +considered +perturbed +Hamiltonian +would +as +. +2 +2 +tot +N N +N +mix +N +H +E +H + + + + + + + + whereAlso, we . is the unperturbed energy +. +2 +N N +E + +determined the expectation values of the Hmix through +11 +12 +21 +22 +mix +mix +mix +mix +H +H +H +H + + + + + + + +2 +N +N + + + + + + + + + += Ei +2 +N +N + + + + + + + + + + +equation and the results have presented in Table 2. To get the results in N+2 bosons space, the + + +9 + +quantum numbers changed as N+2=2k + νs + ν, and therefore, the prediction of regular states +was modified, too. In this stage, we have determined energy spectra of 112Cd nucleus via the +expectation values of i) combination of regular Hamiltonian, HSU(1,1), together Casimir operator +of O(6) dynamical symmetry and ii) total Hamiltonians, which introduced in Eq.(13). We do +this, to show the effect of mixing term individually on the considered states and especially the +intruder ones. Similar to what we have done for regular Hamiltonians, we have examined +different values of cs in our calculation. The best agreement between theoretical predictions +and experimental counterparts, is yield via cs = 0.5 for both SU(1,1) transitional Hamiltonians +in regular, and mixed terms. Also, the κ & κ' values are listed in this table, too which describe +the effect of N, and N+2 bosons spaces, respectively. +Table 2. Energy; spectra of the 112Cd nucleus were determined via the extended method. Parameters of total +Hamiltonians in Eq.(13) are g = 1.14, α = 500, γ = 20.18, δ = 32.99, η = 27.85 and χ = 10.21 (all in keV). Also; +the best agreement is yield by cs = 0.5 in both regular and mixed parts. Similar to Table 1, σ describes the quality +of extraction processes. + +Level k ν νs κ κ' Eexp +  + + +1ˆ +6 +regular +H +C +O + + +tot +H + + + +10 5 0 0 0.99 0.14 0 0 0 + +12 4 2 0 0.96 0.28 617 655 628 + +2 +0 3 2 0 0.91 0.41 1224 1285 1260 + +2 +2 3 4 0 0.81 0.59 1312 1345 1360 + +14 4 2 0 0.87 0.49 1415 1506 1545 + +3 +0 3 3 1 0.79 0.61 1433 1438 1486 + +3 +2 3 4 0 0.76 0.65 1468 1482 1497 + +4 +0 3 2 2 0.64 0.77 1870 1758 1701 + +2 +4 3 4 0 0.37 0.93 1871 1893 1914 + 13 4 2 0 0.81 0.59 2065 2012 2245 + +3 +4 3 3 1 0.61 0.79 2082 2147 2105 + +4 +2 3 2 2 0.57 0.82 2122 2163 2149 + +5 +2 3 1 3 0.52 0.85 2156 2198 2170 + +16 4 2 0 0.74 0.67 2168 2200 2431 + + σ 46.39 22.76 + + A detailed description of the selected energy levels, experimental values together theoretical +predictions of regular, mixed, and total Hamiltonians are presented in Figure 1. + + +10 + + +Figure 1. Energy; spectra of 112Cd nucleus. a); The experimental values, b) theoretical prediction SU(1,1)-based +transitional Hamiltonian, which yield by cs = 0, c) predictions of total Hamiltonian, and finally d) theoretical +predictions of transitional Hamiltonian together O(6) Casimir operator which the two latest are yield by cs = 0.5. + +The extended Hamiltonian, Eq.(13), obviously improves theoretical predictions for energy +spectra of 112Cd. These corrections are apparent for the regular states while for intruder levels, +2 +4  & +3 +2  , this approach completely changes our previous results. The comprehensive process +suggests more exact results for the levels in a ground band compared with the excited bands, +too. Also, the accuracy of this extended formalism decreased for high spin levels, even for J = +6, and one may conclude that we must consider the sdg-IBM version of such comprehensive +models. These results confirm the idea of symmetry mixing and extension of Hamiltonians in +describing intruder structures. On the other hand, for intruder states, the results of column 8 of +Table 2, which yield the expectation values of transitional Hamiltonian together O(6) Casimir +operator, show better agreement. This may relate to the nature of these levels, which have been +reported in Ref.[19]. And also, our results for κ & κ' coefficients verify it. The weight of N and +N+2 bosons space, described by these coefficients, give exciting results about regular and +intruder levels, too. For regular states in the ground band, the N boson space, which +corresponds with the U(5) dynamical limit, has a prominent role, and the impact of this +dynamical limit has reduced when the spin of the considered state increases. For other levels +in the excited bands, the effect of N+2 boson space increased, and the maximum values of κ' +yield for +2 +4  & +3 +2  intruder states. In the next section, we consider the quadrupole transition rates + +1 +a +d +2500- +.61 +6t +-6i +25 : +2§ +6t +三 24 +24 +43 +21 +43 +2↑ +23 +4 +_24 +2000- +31 +42 +31 +43 二 +Is(kev) +4 +04 +4 +42 +-04 +-04 +04 +- 2§ +4t +4 +4i +1500- +25 +2# +2 +level: +E0 +22 : +2 +2 +6 1000 - +2 + 21 +21 + 21 +500 - +F0 +11 + +and examine the effect of considering wave functions as combining two different boson spaces +on theoretical predictions for these quantities. +3. E2 Transition probabilities +Electromagnetic transition rates are known as the most sensitive observable to a nuclear +structure, and their analyses make remarkable data on the mixing of symmetries [19,22,31,32], +shape coexistence [16,19], and partial dynamical symmetry [19,33].The; quadrupole transition +rates are determined by using the following equation: + + +2 +1 +( +2, +) +2 +2 +1 +i +f +f +i +i +B E +L +L +T +E +L + + + + + + (14) +The most general operator within the IBM-1 framework for E2 transition is [25]: + + +2 +2 +2 +( +) +( +) +, +T E +e +s +d +d +s +d +d + + + + + + + + + + + + + + (15) +The selection rules for two terms of this operator are +1 +dn + +  and +0 +dn + + +, respectively. In +this section, we calculated such transition rates of 112Cd nucleus whose experimental data are +available [19] and happened between our considered states, e.g., up to +16 level. We considered +two ways in our calculation, and compared the effect of different bosons spaces on the +theoretical results. In the first one, all of the regular and intruder states have been labeled in the +only N boson space, and therefore, we considered wave functions of Eq.(14) as + are statestion of these he second approach, the wavefun n tI . +, ; +i +N +s +N k +n LM + + +  + + + +described as the combination of N and N+2 bosons spaces, e.g. +2 +' +i +N +N + +  +  + + + +. Also, +we used the κ & κ' values listed in Table 2. The effective charge, e, and the dimensionless +quantity, χ, in Eq.(15) are extracted in comparison with experimental data, too, whose values +are e = 1.857 W.u. and χ = - 0.17. The; prediction of these two approaches have presented in +Table 3. +Table 3. The; experimental quadrupole transition probabilities with theoretical predictions. B(E2)N;is the +theoretical predictions that yielded by using the considered states in N boson space, and B(E2)mix is determined +by using the wave functions as the combination of two N and N+2 boson spaces. + + Transition B(E2)exp B(E2)N B(E2)mix + + +1 +1 +2 +0 + + + + 30.3 36.2 32.5 + +2 +1 +0 +2 + + + + 51.0 49.1 49.6 + + +12 + + +2 +1 +2 +2 + + + + 39.0 42.7 41.3 + +2 +1 +2 +0 + + + + 0.65 0.73 0.69 + +1 +1 +4 +2 + + + + 63.0 71.9 68.8 + +3 +2 +0 +2 + + + + 99.0 113.1 104.8 + +3 +1 +0 +2 + + + + 0.012 0.017 0.014 + +3 +2 +2 +0 + + + + 120.0 138.1 127.6 + +3 +1 +2 +0 + + + + 0.88 1.09 0.97 + +1 +1 +3 +4 + + + + 25.0 30.4 27.2 + +1 +2 +3 +2 + + + + 1.80 2.09 1.97 + +3 +3 +4 +2 + + + + 59.0 68.4 62.7 + +3 +3 +4 +0 + + + + 24.0 29.6 26.5 + +3 +2 +4 +2 + + + + 58.0 76.6 66.2 + +3 +1 +4 +2 + + + + 0.90 1.13 0.97 + +4 +3 +2 +0 + + + + 25.0 31.1 29.4 + +4 +2 +2 +2 + + + + 5.30 6.27 5.88 + +4 +1 +2 +2 + + + + 2.20 2.58 2.41 + +4 +1 +2 +0 + + + + 0.017 0.023 0.020 + +5 +3 +2 +2 + + + + 23.0 29.8 27.2 + +5 +1 +2 +2 + + + + 0.06 0.084 0.072 + +Theoretical results in both approaches suggest good agreement with their experimental +counterparts and this verifies our selection for quantum numbers and also shows the accuracy +of extraction processes. It was evident that when we consider the wave functions as the +combination of states in both N and N+2 bosons spaces, the deviation between theoretical +results and experimental values reduce obviously. The accuracy of these predictions is more +for contraband transitions in comparison with inter band transition in both approaches. Also; +the results of mixed formalism are so apparent for such transitions, which originated from or +ended in intruder states. +The latest subject which we considered in this investigation is the test of a method that consists +of extending wave functions of intruder states to describe transition rates with high exactness. +To; this aim, we used two +3 +1 +2 +0 + + + +and +3 +2 +2 +0 + + + +transitions and considered the state as: + + +13 + +' +3 +1 +2 +2 +0 +0 + + + + + + + + +By using this definition in Eq.(14) and also λ = 0.76 and λ' = 0.65, which are yielded via +extraction in comparison to experimental values, the probability of these transitions are yield +as + + +3 +2 +2,2 +0 +B E + + + += 127.6 and + + +3 +1 +2,2 +0 +B E + + + += 0. 99. One; may conclude the equivalency +these two different methods in the calculation of such quadrupole transition rates due to the +results yield for λ & λ' which are similar to the values of κ & κ' for this intruder state and also +the exact predictions for transition rates. These; results with better predictions of mixed +Hamiltonian for the energy spectra in comparison with the SU(1,1) transitional Hamiltonian +show the necessity to use mixed formalisms for the accurate description of regular and intruder +states and also, transition rates. +4. Conclusions; +In this study, we introduced an extension of SU(1,1)- based Hamiltonian, for a detailed +description of energy levels and quadrupole transition rates of the 112Cd nucleus. This +considered transitional Hamiltonian suggests good predictions only for regular states, but for +the intruders, ones need some corrections. The ability of considered Hamiltonian to coverage +both U(5) and O(6) dynamical limits with standard O(5) sub algebras of them make it possible +to extend this formalism by adding the Casimir operator of O(6) algebra and also mixing term. +We considered the wave functions of both regular and intruder states as a combination in the +N and N+2 boson spaces. The results of mixed formalism show improvement in the predictions +for energy spectra and quadrupole transition rates. We would test the same procedure in the +following research by adding the Casimir operator of U(5) dynamical limit for such nuclei +located near the Z = 82 closed shell, and their intruder states have spherical nature. +Acknowledgment; +This work is supported by the Research Council of the University of Tabriz. +Author contributions; +M. Rastegar, H. Sabri, and A. O. Ezzati performed the initial calculations, analyzed and +interpreted the results, and wrote the main manuscript text. All authors commented on and +reviewed the manuscript. +Competing interests; +The authors declare no competing interests. + + +14 + +References +[1].Pan, Feng, Xin Zhang, and J. P. Draayer. "Algebraic solutions of an sl-boson system in the U (2l+ 1)⟷ O (2l+ +2) transitional region." Journal of Physics A: Mathematical and General 35, no. 33 (2002): 7173. +[2]-Bohr, Niels. On the transmutation of atomic nuclei by the impact of material particles. 1937. +[3]. Heyde, Kris, and John L. Wood. "Publisher’s Note: Shape coexistence in atomic nuclei [Rev. Mod. Phys. 83, +1467 (2011)]." Reviews of Modern Physics 83, no. 4 (2011): 1655. +[4]. Heyde, K., P. Van Isacker, M. Waroquier, J. L. Wood, and R. A. Meyer. "Coexistence in odd-mass nuclei." +Physics Reports 102, no. 5-6 (1983): 291-393. +[5]. Cejnar, Pavel, Jan Jolie, and Richard F. Casten. "Quantum phase transitions in the shapes of atomic nuclei." +Reviews of Modern Physics 82, no. 3 (2010): 2155. +[6]. Rowe, D. J. 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Chinese Physics C 44, no. 6 (2020): 064102. + diff --git a/OtE0T4oBgHgl3EQfjwFX/content/tmp_files/load_file.txt b/OtE0T4oBgHgl3EQfjwFX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..de7c5757e0445821e98b32022555632efbddc542 --- /dev/null +++ b/OtE0T4oBgHgl3EQfjwFX/content/tmp_files/load_file.txt @@ -0,0 +1,812 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf,len=811 +page_content='The exact description of intruder states in 112Cd nucleus by using a mixing formalism based on SU(1,1( transitional Hamiltonian and O(6)–Casimir operator M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Rastegar*, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Sabri ,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Ezzati Department of Physics, University of Tabriz, 51664 Tabriz, Iran Corresponding author e-mail: m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='rastgar@tabrizu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='ir 2 Abstract;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' In this paper, we used a transitional Hamiltonian which has U(5)↔O(6) transition to improve theoretical predictions for energy spectra and quadrupole transition rates of 112Cd nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' To this aim, the transitional Hamiltonian in the affine SU(1,1) algebra has been extended by adding the O(6)-Casimir operator and mixing Hamiltonian to increase exactness in the description of 24 \uf02b and 32 \uf02b intruder levels of this nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' We also considered the wave functions of both regular and intruder states, as a combination in the N and N+2 boson spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The results confirm the advantages of using such mixing approaches and describing the energy and transition rates with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Keywords: Interacting boson model (IBM), particle-hole excitation, intruder levels, transitional Hamiltonian, quadrupole transition rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Introduction One of the genius nuclei that are in the transitional between vibrational and axial symmetry regions are 110,112,114Cd but in this paper, we want to test just 112Cd nucleus because it seems great exhibiting for our method and also Lehman in the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='emphasize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Also in Ref [35] mentioned about the particle-hole excitations in 108Cd by the intruder states that just worked by Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' There is a novelty way in this paper that it is work by expansion wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The intruder levels and their observation via experimental techniques, with descriptions of these levels by the theoretical framework, are the subject of different studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The work of Bohr and Kalkar in nuclear physics has exposed the intention of deformed shapes and the manifestation of disparate shapes in a given nucleus [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' This idea has been developed by other original studies, such as [2-5], which used different shapes in a nucleus for a complete description of its observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=" Various models based on the shell model's concepts, collective geometric model, models based on the mean-field approach with algebraic frameworks are used for such investigations." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' On the other hand, the multi particles multi holes (mp-mh) is another exclusive method which first proposed in the shell model framework and then expanded in the algebraic models such as the interacting boson model (IBM) to describe different intruder levels in nuclei that are located near the closed proton and neutron shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The idea of expanding the ordinary 2p-2h intruder excitations into two pairs of extra nucleons (two 3 bosons) and studying their mixing with the regular configurations that first Dual and Barrett presented this approach [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' It has been widely reported that intruder states have been exposed to low excitation energies [4-10,11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' mp-mh excitations cannot be compounded easily in complete large-scale shell-model studies because of the extensive extents of the model spaces involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The new label has been introduced to represent the symmetries connecting particle and hole bosons within the IBM configuration mixing calculations [13-17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' In this approach, the effect of such excitations is expressed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' as different terms of regular Hamiltonian and, therefore;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' a combination of regular and mixing Hamiltonians e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' other symmetries, improve theoretical predictions for both regular and intruder states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' A transitional Hamiltonian between different symmetry limits of IBM provides a similar combination of different symmetries, and this idea is the main subject of this investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' A detailed investigation of both regular and intruder levels of the Cd isotopic chain has been done by Lehman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' They have claimed that the vibrational characteristics of 112Cd portray by the regular configuration subtending of N bosons, s, and d bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' As a result of the more deformation nature of the intruder configuration, there are two extra bosons involved, N+2, which takes account of a 2p-2h excitation within the closed shells [18-19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' They showed that the 2 4 \uf02b and 3 2 \uf02b are intruder states for U(5) algebra, and as anticipated, this limit is too narrow to duplicate the whole set of empirical data, as some low-lying states are outside the model space [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' To improve the contract between theoretical and experimental results, a subsequent configuration called the intruder configuration had been required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Therefore, they have combined the regular and intruder configurations, which correspond with U(5) and O(6) dynamical limits of IBM, respectively, for N and N+2 bosons spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' In this investigation, we extend a similar combination by using a transitional Hamiltonian between U(5) and SO(6) limits which are based on affine SU(1,1) algebra [16-17] as a regular Hamiltonian in the N boson space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' In recent studies, this Hamiltonian was used to explain the coexistence of deformed shapes by using the two control parameters of transitional Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Then, this Hamiltonian mixed with the O(6) Casimir operator in the N+2 bosons space to describe intruder levels of 112Cd nucleus with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' We also considered the advantages of this mixed formalism in describing of different quadrupole transition rates of this nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The regular part of Hamiltonian (unperturbed representation): For the nuclei in the vicinity of the proton closed shell at Z = 50, such as Cd isotopic chain, the regular configuration can be exposed by U(5) algebra in the IBM model, and the intruder configuration gives away the O(6) symmetry [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Intruding, the construction owning to the particle-hole excitation amidst the closed shells is a well-known particularity in the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' This configuration is a new implement for the exposition of the level scheme of nuclear that has not been predicted in one algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' In;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' cases where the Hamiltonian could be written with the help of the limits of the Casten triangle, the Hamiltonian could be written with invariants operators of two of the three chains [24,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' One of these transitions is U(5)↔O(6) transition which is used to describe nuclei between vibrational and gamma-unstable limits, nuclei near the closed shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' In the framework of IBM, the even-even Cd isotopes are good candidates to perform the such a transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' This means one may use such transitional Hamiltonian to investigate energy spectra and transition rates of nuclei in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The advantages of SU(1,1)-based transitional Hamiltonian in comparison with other algebraic approaches which are illustrated in detail in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='[17-18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Encourage us to focus only on this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' We used this SU(1,1) transitional Hamiltonian by the definition of [N] boson presentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' details, about this transitional Hamiltonian, are available in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' [17-18,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Here, we briefly introduce the main concepts and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The SU(1,1) algebra has been constructed by , 0, S\uf06e \uf06e \uf03d \uf0b1 operators in which the generators d \uf02d boson pairing algebra are: † † 0 † † 1 1 1 ( ) ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' ) , ( ) ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' ( ) ( ) (1) 2 2 4 S d d d S d d d S d d d d d \uf06e \uf06e \uf06e \uf06e \uf06e \uf02b \uf02d \uf03d \uf03d \uf03d \uf02b \uf0e5 And these operators for s \uf02d boson pairing algebra define as: †2 2 0 † † 1 1 1 ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' ( ) ( ) (2) 2 2 4 S s s S s s S s s s ss \uf02b \uf02d \uf03d \uf03d \uf03d \uf02b The infinite-dimensional SU(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='1) algebra has been generated by using [17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='18]: 2 1 2 1 0 2 0 2 0 ( ) ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' ( ) ( ) (3) n n n n n s d n s d S c S s c S d S c S s c S d \uf0b1 \uf02b \uf0b1 \uf02b \uf0b1 \uf03d \uf02b \uf03d \uf02b Where cs and cd are real parameters and n can be 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' \uf0b1 \uf0b1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The commutation relations between these operators are: 5 0 0 1 [ , ] , [ , ] 2 (4) m n m n m n m n S S S S S S \uf0b1 \uf0b1 \uf02b \uf02d \uf02b \uf02b \uf02b \uf03d \uf0b1 \uf03d \uf02d Then, affine Lie algebra SU(1,1) without central extension has been constructed by using these generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Casimir operator is 0 0 2ˆ ( 1) C S S S \uf0b1 \uf03d \uf02d \uf02d , too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The IBM-1 Hamiltonian, without distinguishing between proton and neutrons, in this formalism, has written as: 0 (1,1) 0 0 1 2ˆ ˆ ˆ ε [( (5) (3)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' regular SU H H gS S S C SO SO \uf067 \uf064 \uf02b \uf02d \uf0ba \uf03d \uf02b \uf02b \uf02b ( ε , γ , δ and g are real parameters) (5) Cs = Cd corresponds with the SO(6) limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Also, the U(5) limit has been described by the cs = 0 requirements, and finally, cs ≠ cd has been used for the explanation of the U(5)↔O(6) transitional region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' We present the regular or unperturbed states of SU(1,1) algebra by with ix number parameters c \uf02d n terms of unknown , dwhich define , ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' s N k n LM \uf06e \uf06e \uf044 as:, 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=', i k \uf03d 1 2 1 2 1 2 1 2 , ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' , (6) k k k i n n n s n n n k n n n n Z N k n LM a a a x x x S S S lw \uf06e \uf06e \uf02b \uf02b \uf02b \uf044 \uf0ce \uf03d \uf0e5 k relate to the total number of boson N as N = 2k + ν + νs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The analytical behavior of wave functions allows us to consider ix near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Now, the wave functions of SU(1,1) algebra are yielded by using the commutation relations between the generators as: 1 2 , ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' , (7) k s x x x N k n LM NS S S lw \uf06e \uf06e \uf02b \uf02b \uf02b \uf044 \uf03d Where N is the normalization factor and 2 2 ( ) ( ) , (8) 1 1 i s d x s i d i c c S S s S d c x c x \uf02b \uf02b \uf02b \uf03d \uf02b \uf02d \uf02d Also, we defined the xi values via the following set of equations: 2 2 2 2 1 5 ( ) ( ) ε 2 2 2 , 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=', (9) 1 1 s s d i j i s i d i i j gc gc for i k x c x c x x x \uf06e \uf06e \uf0b9 \uf02b \uf02b \uf03d \uf02b \uf03d \uf02d \uf02d \uf02d \uf0e5 The eigenvalues of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='5, regular Hamiltonian by 0p-0h excitation, are determined as: \uf028 \uf029 (1,1) ˆ , ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' , ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' (10) k s SU s E N k n LM H N k n LM \uf06e \uf06e \uf06e \uf06e \uf044 \uf044 \uf03d The final form is: 6 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 0 1 0 2 2 1 1 3 1 1 1 5 and 2 2 2 k k k k s d i i E h L L c c h s x \uf067\uf06e \uf06e \uf064 \uf061 \uf061 \uf06e \uf06e \uf03d \uf03d \uf02b \uf02b \uf02b \uf02b \uf02b \uf04c \uf0e9 \uf0f9 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf04c \uf03d \uf02b \uf02b \uf02b \uf03d \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0ea \uf0fa \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0eb \uf0fb \uf0e5 (11) Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' we must extract the parameters of this equation in comparison with experimental data for energy levels of 112Cd nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' To this aim firstly, we selected, ε g \uf065 \uf03d , g = 1 keV, 1 s d c c c \uf03d \uf0a3 similar to adn alautc values of ase wit rofhc 1 i \uf03d ) have solved for8then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' ( 2 yi d i c x \uf03d and what have done in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' After getting the roots for each level, and m identical to what that have done in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' \uf06e and s \uf06e combination into h(k), we can use some extraction process such as least square fitting are other global methods but to get the exact results we used the roots of equations by beth-anthats methods, we follow the method that, used the least square fit in Matlab software to extract δ and γ [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' These processes, have been repeated by using the different values of c and ε to reduce the difference between theoretical predictions for different regular and intruder energy levels and experimental counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' We changed the cs values between 0 and 1 limits with Δcs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='1 step length;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' determined all roots, parameters, and energy values for each case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' some of these results have presented in Table1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='We also used the \uf028 \uf029 2 exp 2 1 Th i E E N \uf073 \uf03d \uf02d \uf0e5 , N, the number of considered levels, as the quality measure for our extracting procedure, which their values for different cs are present in Table 1, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Table1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Regular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' and intruder energy levels of 112Cd nucleus have been determined by using different cs values with related quantum numbers in the framework of SU(1,1) algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The experimental data are taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' σ describes the quality of extraction processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' All;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' of the energy values are in keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Level k ν νs Eexp Eth(cs = 0) Eth(cs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='2) Eth(cs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='4) Eth(cs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='6) Eth(cs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='Eth(cs = 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='10\uf02b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='2454 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='2574 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='2679 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='2321 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='2376 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='16 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='16 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='99 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='11 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='25 321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='66 Our results show good predictions of the transitional Hamiltonian, independent of cs values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' We examine different values of cs and the best agreement between theoretical and experimental counterparts found in cs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='This confirms our idea of using such transitional formalism as a regular part of this nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' These results are in agreement with the predictions of Jolie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' [31-34] and Heyde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' [31-32], which showed the advantages of U(5) dynamical symmetry in the description of the regular part in the energy spectra of the Cd isotopic chain, respectively shell model and a combination of U(5) and O(6) dynamical limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Also, only for 2 4 \uf02b and 3 2 \uf02b as intruder levels, the distance between theoretical predictions and experimental values are apparent, and for other levels in ground and excited bands, this formalism makes appropriate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Another;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' point in this table is the better results of the cs = 0 case, which, as we have expected for the N bosons space, the U(5) limit excepted for such space and also described regular states with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' This result confirms the predictions of Lehman et al in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' [7,19], which used the Hamiltonian of U(5) limit for the regular states of this nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' In the following, we add new terms to our regular Hamiltonian and also suppose the N+2 bosons space in determining of the effects of these new terms to make an accurate description of intruder levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' This idea, adding the O(6) limit as a perturbation to the regular U(5) Hamiltonian, is similar to the method that Heyde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' [31,32], have proposed because both groups have a standard O(5) subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Additional terms The affine SU(1,1) lie algebra, has been used here to provide a relevantly simple solvable pairing model that incorporates the mixing of two-particle and two-hole configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' This 8 transitional Hamiltonian in our calculations in the previous subsection works on the states of N bosons space described as , ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' s N k n LM \uf06e \uf06e \uf044 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Now, we introduce the method based on the extension of this transitional Hamiltonian by adding the Casimir operator of O(6) dynamical limit and also carrying out the calculation in the N+2 space, which the total wave function with the 2, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' s N k n LM \uf06e \uf06e \uf044 \uf02b type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' In this way, we followed the method developed by Lehman and Jolie, with the nature of 2 4 \uf02b and 3 2 \uf02b states, as have been predicted by Heyde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' [22,31,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Rowe, in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' [22,24] has claimed that ˆ ˆ S S \uf02b \uf02d the operator in the framework of SU(1,1) algebra is the Casimir operator of O(6) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' We extended our approach by adding the ˆ ˆ S S \uf02b \uf02d term to affine SU(1,1) algebra as 2p-2h excitation, similar to the method done by Lehman in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' In the framework of U(6) algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Another term that must add to our extended Hamiltonian is the mixing term which combines both N and N+2 bosons spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' To this aim, we used the usual mixing Hamiltonian, which is defined in different papers as: 0 0 ˆ [ ] [ ] mix H s s s s d d d d \uf068 \uf063 \uf02b \uf02b \uf02b \uf02b \uf03d \uf0b4 \uf02b \uf0b4 \uf02b \uf0b4 \uf02b \uf0b4 (12) We also used that our considered transitional Hamiltonian connects both U(5) and O(6) dynamical limits by the variation of a control parameter of transitional Hamiltonian and, therefore;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' has the same role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The Hamiltonian to illustrate mixing could be written as follows: \uf028 \uf029 \uf028 \uf029 (1,1) 2 1 2 (1,1) 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ( ) ( 6 ) ( ) tot N SU N N N N SU mix N H P H P P gC O gS S P P H H P \uf02b \uf02d \uf02b \uf02b \uf02b \uf03d \uf02b \uf0ba \uf02b \uf02b (13) The;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' PN is the projection operator that represents the N-boson subspaces, while PN+2 projects the subspaces to N + 2 bosons [18-19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' To determine the energy levels for these new extended Hamiltonian, we used again Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=" (10), where the considered i \uf079 states to get eigenvalues are yielded by combining the two wave functions as: 2 ' i N N \uf079 \uf06b \uf079 \uf06b \uf079 \uf02b \uf03d \uf02b ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The eigenvalues of this considered perturbed Hamiltonian would as .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' 2 2 tot N N N mix N H E H \uf064 \uf079 \uf079 \uf02b \uf02b \uf03d \uf02b whereAlso, we .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' is the unperturbed energy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' 2 N N E\uf064 \uf02b determined the expectation values of the Hmix through 11 12 21 22 mix mix mix mix H H H H \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e8 \uf0f8 2 N N \uf079 \uf079 \uf02b \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e8 \uf0f8 = Ei 2 N N \uf079 \uf079 \uf02b \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e8 \uf0f8 equation and the results have presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' To get the results in N+2 bosons space, the 9 quantum numbers changed as N+2=2k + νs + ν, and therefore, the prediction of regular states was modified, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' In this stage, we have determined energy spectra of 112Cd nucleus via the expectation values of i) combination of regular Hamiltonian, HSU(1,1), together Casimir operator of O(6) dynamical symmetry and ii) total Hamiltonians, which introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='(13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' We do this, to show the effect of mixing term individually on the considered states and especially the intruder ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Similar to what we have done for regular Hamiltonians, we have examined different values of cs in our calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The best agreement between theoretical predictions and experimental counterparts, is yield via cs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='5 for both SU(1,1) transitional Hamiltonians in regular, and mixed terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=" Also, the κ & κ' values are listed in this table, too which describe the effect of N, and N+2 bosons spaces, respectively." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' spectra of the 112Cd nucleus were determined via the extended method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Parameters of total Hamiltonians in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' (13) are g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='14, α = 500, γ = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='18, δ = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='99, η = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='85 and χ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='21 (all in keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Also;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' the best agreement is yield by cs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='5 in both regular and mixed parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Similar to Table 1, σ describes the quality of extraction processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=" Level k ν νs κ κ' Eexp \uf028 \uf029 \uf028 \uf029 1ˆ 6 regular H C O \uf02b tot H 10\uf02b 5 0 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='14 0 0 0 12\uf02b 4 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='28 617 655 628 2 0\uf02b 3 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='41 1224 1285 1260 2 2\uf02b 3 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='59 1312 1345 1360 14\uf02b 4 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='49 1415 1506 1545 3 0\uf02b 3 3 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='61 1433 1438 1486 3 2\uf02b 3 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='65 1468 1482 1497 4 0\uf02b 3 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='77 1870 1758 1701 2 4\uf02b 3 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='93 1871 1893 1914 13\uf02b 4 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='59 2065 2012 2245 3 4\uf02b 3 3 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='79 2082 2147 2105 4 2\uf02b 3 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='82 2122 2163 2149 5 2\uf02b 3 1 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='85 2156 2198 2170 16\uf02b 4 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='67 2168 2200 2431 σ 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='39 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='76 A detailed description of the selected energy levels, experimental values together theoretical predictions of regular, mixed, and total Hamiltonians are presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' 10 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' spectra of 112Cd nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The experimental values, b) theoretical prediction SU(1,1)-based transitional Hamiltonian, which yield by cs = 0, c) predictions of total Hamiltonian, and finally d) theoretical predictions of transitional Hamiltonian together O(6) Casimir operator which the two latest are yield by cs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The extended Hamiltonian, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' (13), obviously improves theoretical predictions for energy spectra of 112Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' These corrections are apparent for the regular states while for intruder levels, 2 4 \uf02b & 3 2 \uf02b , this approach completely changes our previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The comprehensive process suggests more exact results for the levels in a ground band compared with the excited bands, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Also, the accuracy of this extended formalism decreased for high spin levels, even for J = 6, and one may conclude that we must consider the sdg-IBM version of such comprehensive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' These results confirm the idea of symmetry mixing and extension of Hamiltonians in describing intruder structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' On the other hand, for intruder states, the results of column 8 of Table 2, which yield the expectation values of transitional Hamiltonian together O(6) Casimir operator, show better agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' This may relate to the nature of these levels, which have been reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=" And also, our results for κ & κ' coefficients verify it." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The weight of N and N+2 bosons space, described by these coefficients, give exciting results about regular and intruder levels, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' For regular states in the ground band, the N boson space, which corresponds with the U(5) dynamical limit, has a prominent role, and the impact of this dynamical limit has reduced when the spin of the considered state increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=" For other levels in the excited bands, the effect of N+2 boson space increased, and the maximum values of κ' yield for 2 4 \uf02b & 3 2 \uf02b intruder states." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' In the next section, we consider the quadrupole transition rates 1 a d 2500- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='61 6t 6i 25 : 2§ 6t 三 24 24 43 21 43 2↑ 23 4 _24 2000- 31 42 31 43 二 Is(kev) 4 04 4 42 04 04 04 2§ 4t 4 4i 1500- 25 2# 2 level: E0 22 : 2 2 6 1000 - 2 21 21 21 500 - F0 11 and examine the effect of considering wave functions as combining two different boson spaces on theoretical predictions for these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' E2 Transition probabilities Electromagnetic transition rates are known as the most sensitive observable to a nuclear structure, and their analyses make remarkable data on the mixing of symmetries [19,22,31,32], shape coexistence [16,19], and partial dynamical symmetry [19,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='The;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' quadrupole transition rates are determined by using the following equation: \uf028 \uf029 2 1 ( 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' ) 2 2 1 i f f i i B E L L T E L \uf079 \uf079 \uf0ae \uf03d \uf02b (14) The most general operator within the IBM-1 framework for E2 transition is [25]: \uf028 \uf029 2 2 2 ( ) ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' T E e s d d s d d \uf063 \uf02b \uf02b \uf02b \uf0e9 \uf0f9 \uf03d \uf0b4 \uf02b \uf0b4 \uf02b \uf0b4 \uf0eb \uf0fb (15) The selection rules for two terms of this operator are 1 dn \uf044 \uf03d \uf0b1 and 0 dn \uf044 \uf03d ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' In this section, we calculated such transition rates of 112Cd nucleus whose experimental data are available [19] and happened between our considered states, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=', up to 16\uf02b level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' We considered two ways in our calculation, and compared the effect of different bosons spaces on the theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' In the first one, all of the regular and intruder states have been labeled in the only N boson space, and therefore, we considered wave functions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' (14) as are statestion of these he second approach, the wavefun n tI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' , ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' i N s N k n LM \uf079 \uf079 \uf06e \uf06e \uf044 \uf03d \uf0ba described as the combination of N and N+2 bosons spaces, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=" 2 ' i N N \uf079 \uf06b \uf079 \uf06b \uf079 \uf02b \uf03d \uf02b ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=" Also, we used the κ & κ' values listed in Table 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The effective charge, e, and the dimensionless quantity, χ, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' (15) are extracted in comparison with experimental data, too, whose values are e = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='857 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' and χ = - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' prediction of these two approaches have presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' experimental quadrupole transition probabilities with theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' B(E2)N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='is the theoretical predictions that yielded by using the considered states in N boson space, and B(E2)mix is determined by using the wave functions as the combination of two N and N+2 boson spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Transition B(E2)exp B(E2)N B(E2)mix 1 1 2 0 \uf02b \uf02b \uf0ae 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='2 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='5 2 1 0 2 \uf02b \uf02b \uf0ae 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='1 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='6 12 2 1 2 2 \uf02b \uf02b \uf0ae 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='7 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='3 2 1 2 0 \uf02b \uf02b \uf0ae 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} 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+page_content='30 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='88 4 1 2 2 \uf02b \uf02b \uf0ae 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='58 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='41 4 1 2 0 \uf02b \uf02b \uf0ae 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='020 5 3 2 2 \uf02b \uf02b \uf0ae 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='8 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='2 5 1 2 2 \uf02b \uf02b \uf0ae 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='072 Theoretical results in both approaches suggest good agreement with their experimental counterparts and this verifies our selection for quantum numbers and also shows the accuracy of extraction processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' It was evident that when we consider the wave functions as the combination of states in both N and N+2 bosons spaces, the deviation between theoretical results and experimental values reduce obviously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The accuracy of these predictions is more for contraband transitions in comparison with inter band transition in both approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Also;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' the results of mixed formalism are so apparent for such transitions, which originated from or ended in intruder states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The latest subject which we considered in this investigation is the test of a method that consists of extending wave functions of intruder states to describe transition rates with high exactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' To;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=" this aim, we used two 3 1 2 0 \uf02b \uf02b \uf0ae and 3 2 2 0 \uf02b \uf02b \uf0ae transitions and considered the state as: 13 ' 3 1 2 2 0 0 \uf06c \uf06c \uf02b \uf02b \uf02b \uf03d \uf02b By using this definition in Eq." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' (14) and also λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content="76 and λ' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='65, which are yielded via extraction in comparison to experimental values, the probability of these transitions are yield as \uf028 \uf029 3 2 2,2 0 B E \uf02b \uf02b \uf0ae = 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='6 and \uf028 \uf029 3 1 2,2 0 B E \uf02b \uf02b \uf0ae = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' One;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=" may conclude the equivalency these two different methods in the calculation of such quadrupole transition rates due to the results yield for λ & λ' which are similar to the values of κ & κ' for this intruder state and also the exact predictions for transition rates." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' These;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' results with better predictions of mixed Hamiltonian for the energy spectra in comparison with the SU(1,1) transitional Hamiltonian show the necessity to use mixed formalisms for the accurate description of regular and intruder states and also, transition rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Conclusions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' In this study, we introduced an extension of SU(1,1)- based Hamiltonian, for a detailed description of energy levels and quadrupole transition rates of the 112Cd nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' This considered transitional Hamiltonian suggests good predictions only for regular states, but for the intruders, ones need some corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The ability of considered Hamiltonian to coverage both U(5) and O(6) dynamical limits with standard O(5) sub algebras of them make it possible to extend this formalism by adding the Casimir operator of O(6) algebra and also mixing term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' We considered the wave functions of both regular and intruder states as a combination in the N and N+2 boson spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The results of mixed formalism show improvement in the predictions for energy spectra and quadrupole transition rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' We would test the same procedure in the following research by adding the Casimir operator of U(5) dynamical limit for such nuclei located near the Z = 82 closed shell, and their intruder states have spherical nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Acknowledgment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' This work is supported by the Research Council of the University of Tabriz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Author contributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Rastegar, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Sabri, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Ezzati performed the initial calculations, analyzed and interpreted the results, and wrote the main manuscript text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' All authors commented on and reviewed the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Competing interests;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' 14 References [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='Pan, Feng, Xin Zhang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Draayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' "Algebraic solutions of an sl-boson system in the U (2l+ 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nuclei: Application to even-even Dy nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='" Physical Review C 29, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' 4 (1984): 1428 [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Heyde, Kristiaan, Christiaan De Coster, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Wood, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Jolie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' "Proton 2p-2h intruder excitations and the modified vibrational intensity and selection rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='" Physical Review C 46, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' 5 (1992): 2113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Dai, Lianrong, Feng Pan, Ziwei Feng, Yu Zhang, Sai Cui, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' Draayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' "Exact solution of U (5)– O (6) transitional description in interacting boson model with two-particle and two-hole configuration mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content='" Chinese Physics C 44, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} +page_content=' 6 (2020): 064102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfjwFX/content/2301.02463v1.pdf'} diff --git a/P9E0T4oBgHgl3EQf1AL3/content/2301.02695v1.pdf b/P9E0T4oBgHgl3EQf1AL3/content/2301.02695v1.pdf new file mode 100644 index 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The official published version will appear in [2023] Public Law, +forthcoming (published by Sweet & Maxwell) and made available in digital format on subscription from Westlaw UK. + +How do ‘technical’ design-choices made when building algorithmic decision- +making tools for criminal justice authorities create constitutional dangers? + +by Karen Yeunga and Adam Harkensb + +Part II +1. Introduction + +The high-profile failure of automated digital decision tools by public authorities, particularly those that utilise +some form of machine learning (ML), including the ‘robo-debt’ scandal in Australia1and the Dutch childcare +benefits scandal,2 illustrate the scale and seriousness of the hardship and injustice that digital ‘solutions’ in +government can produce. Reflecting on these systems, the UN’s previous Special Rapporteur on Extreme +Poverty, Philip Alston has warned of the imminent dangers of ‘digital dystopia’3 highlighting the urgent need +for safeguards. Although established public law principles could be mobilised to prevent mistakes and failures, +they are yet to be effectively and systematically operationalised in the development, implementation, and +oversight of public sector algorithmic tools. This two-part paper focuses on digital tools used by criminal justice +authorities that purport assess the ‘risk’ posed by specific individuals to inform how they should be treated, +although much of our analysis has wider applicability to the public sector more generally. In Part I, we showed +how ML-based automated digital decision-making and support tools (‘algorithmic tools’) are conventionally +built and implemented without reference to their larger context of application. We argued that, despite the +‘rights-critical’ nature of criminal justice decisions,4 algorithmic tools for public sector use are conventionally +developed in ignorance of public law principles and the legal duties to which they give rise. For technical +developers, these contextual considerations are conventionally regarded as irrelevant ‘noise’, informed by what +we call a ‘contextual detachment mindset’.5 As a result, vital institutional safeguards against the arbitrary or +otherwise unjust exercise of power by public authorities are being circumvented, substantially enhancing the +likelihood that public powers may be exercised unlawfully, creating injustice that is in practice difficult to detect +and almost impossible to challenge. In this Part II, we demonstrate more precisely how choices made by +technical developers during the design, construction and implementation of algorithmic tools implicate several +legal duties that apply to the exercise of the decision-making authority that those tools inform, including English +administrative law doctrine, human rights protections set out in the European Convention on Human Rights + +a Professorial Fellow in Law, Ethics and Informatics, Birmingham Law School and School of Computer Science, +University of Birmingham. We gratefully acknowledge funding support form VW Stiftung, Grant No: 19-0087 (2019- +2023) and for helpful feedback by Emma Ahmed-Rengers (particularly in comparing conventional statistics with ML +approaches), Reuben Binns, Mireille Hildebrandt, Tobias Krafft, Winston Maxwell, Leandro Minku, Johannes Schmees, +Georg Wenzelberger and Katharina Zweig. Karen Yeung drafted the text and devised the analytical framework, +argument, paper structure and narrative. Adam Harkens undertook the in-depth case-studies and background research to +the legal, scholarly and contextual detail supporting Yeung’s arguments and acted as a critical sounding board for her +ideas. An earlier version was presented by Karen Yeung to the Norwegian Association for Computers and the Law, The +Knut Selmer Memorial Lecture, 23 November 2020 (Oslo). +b Post-doctoral Research Fellow, Birmingham Law School. +1 P. Henman, “Administrative justice in a digital world: Challenges and solutions” in J. Tomlinson, R. Thomas, M. Hertogh +and R. Kirkham (eds.) The Oxford Handbook of Administrative Justice (Oxford: OUP, 2021). +2 Melissa Heikilla, “Dutch scandal serves as a warning for Europe over risks of using algorithms” (29 March 2022, Politico +EU), https://www.politico.eu/article/dutch-scandal-serves-as-a-warning-for-europe-over-risks-of-using-algorithms. +3 Philip Alston, ‘Report of the Special Rapporteur on Extreme Poverty and Human Rights’ A/74/48037 (New York: United +Nations, 2019). +4 i.e., such decisions may interfere with the legal and fundamental rights of affected individuals: see Part I section 3. +5 See K.Yeung and A. Harkens, “How do ‘technical’ design-choices made when building algorithmic decision-making +tools for criminal justice authorities create constitutional dangers? Part I” [2022] Public Law, forthcoming at section 2.2. + + + + + + +2 +(ECHR) and incorporated by the Human Rights Act (HRA), data protection laws arising under that Data +Protection Act 2018, and the so-called Public Sector Equality Duty.5 In so doing, we seek to move beyond +existing academic inquiry that has tended to be rather general and abstract. This high-level of generality is +problematic because, as legal scholars commissioned by the Administrative Conference of the United States +(ACUS) to review federal agency use of algorithmic tools have observed: + +“…much, if not most, of the hard work regulating algorithmic governance tools will come not in the constitutional +clouds, but rather in the streets of administrative law.”6 + +We begin with a brief account of three algorithmic tools that purport to assess the ‘risk’ posed by individuals +(‘i-RATS’) currently (or until recently) in use by criminal justice authorities: two tools used in England - the +London Gangs Matrix and the Durham Constabulary’s HART tool, and the SyRI tool formerly used in the +Netherlands. We then examine the intersection between public law and data science perspectives by adopting +the lens of ‘algorithmic regulation’, drawing selectively from these three i-RATs, to demonstrate how particular +abstraction decisions involved in algorithmic model-building directly implicate constitutional principles at each +stage of the development process yet are conventionally and routinely ignored. We argue that algorithmic tool- +developers, and the authorities who commission and implement them, have failed to recognise, or understand, +the constitutional and legal implications of these technical choices. Hence, algorithmic tools are being employed +by criminal justice authorities in ways that unjustifiably violate constitutional principles and the specific legal +duties to which they give rise, significantly enhancing the risk and magnitude of injustice and abuses of power +that can arise from their use. + +The third and fourth sections consider the implications of our analysis. We suggest that constitutional principles +should be mandatory requirements forming an essential part of the design-brief which those who build +algorithmic tools to inform criminal justice decision-making must adhere to. However, because technical +developers cannot be expected to understand nor properly apply public law principles and duties, we argue that +they must collaborate closely with legal experts when deciding whether to deploy these tools for specific +criminal justice purposes, and if justified, to ensure that they are configured in a manner that is demonstrably +compliant with public law principles (including respect for human rights) and all applicable legal duties +throughout the tool-building process. If such compliance cannot be demonstrated, they should not be used. +Given that lawyers unfamiliar with algorithmic model-building may struggle to understand how such principles +and duties are implicated in technical design-choices, we also offer several practical recommendations, +highlighting several ‘detachment practices’ (that is, practices in which legally and constitutionally relevant +matters are conventionally ignored by technical developers) that must be avoided when algorithmic tools are +developed for use by criminal justice decision-makers. Finally, we outline a series of urgently-needed +systematic legal reforms to help establish and maintain public trust in criminal justice decision-making in an +increasingly algorithmic age, followed by a brief conclusion. + +2. +Three algorithmic risk assessment tools used to categorise and ‘flag’ individuals + +We begin with a brief account of three i-RATS used in criminal justice settings based on knowledge gleaned +largely from publicly-available documents: The London Gangs Matrix, the Durham ‘HART’ tool, and the Dutch +SyRI tool. Although their technical dimensions and intended purposes vary significantly (to reduce gang +violence, to enhance the effectiveness of offender ‘rehabilitation’, and more efficient identification of social +welfare fraudsters respectively), they all produce an algorithmically generated assessment of an individual’s +‘risk’ for use by front-line decision-makers when deciding what action to take against those individuals. It is +important to emphasise, however, that their effectiveness in achieving their claimed criminal justice purposes +has not been systematically evaluated, let alone proven, and this, as we shall see, is a matter of considerable +constitutional importance given that they entail prima face interferences with human rights.8 + + +5 Equality Act 2010, s.149. +6 David Freeman Engstrom et al., “Government by algorithm: Artificial intelligence in federal administrative agencies” +(Report submitted to the Administrative Conference of the United States, 2020), 76. +8 E.g., consider the ‘Waterproof’ project referred to in the SyRI case, discussed in section 3.1 of Part I. + + + + + + +3 +2.1 +The London Gangs Matrix + +The Gangs Matrix was created by the London Metropolitan Police Service (MPS) following the London riots +in August 2011 to help reduce ‘street-focused’ gang violence. It provides police officers, via their internet- +enabled smart devices, with a dynamic digital dashboard and database that lists individuals identified as +potentially ‘at risk’ of involvement in gang violence (called ‘Gang Nominals’) together with a ‘harm score’ +generated via an algorithmic assessment tool believed to be created using ML techniques.9 The MPS claims that +the Gangs Matrix (a) helps police officers identify and assess in real-time the ‘risk of violent re-offending’ of a +Gang Nominal by classifying them as ‘high’ (red), ‘medium’ (amber) or ‘low’ (green); and (b) helps allocate +MPS enforcement resources, prioritising those deemed most dangerous while others are diverted and offered +alternative support. Individuals can be included on the Matrix if identified as a potential gang member by a +police officer or partner agency representative (including local councils, schools, and health services) and this +has been corroborated by ‘reliable intelligence from more than one source.’ Intelligence deemed ‘reliable’ could +include the fact that the individual has been observed associating with, being related to, or subject to a stop and +search report together with a Gang Nominal, or even sharing social media content that refers to a specific gang. +Neither details of the model used to calculate the ‘harm score’ nor the data used to create it have been published, +although the MPS states that factors contributing to the score include previous history of violence (weighted by +seriousness of offence), violence or weapons intelligence in the previous six months, judgments of a local gang +unit intelligence manager, and partner organisations’ assessments. How these factors are weighted or utilised to +calculate an individual’s harm score is not publicly known: the MPS merely state that it is based on a ‘complex +scoring system’.7 Individuals listed on the Matrix may be subjected to heightened police surveillance, often +leading to pre-emptive stop and search, arrest for minor offences, anti-social behaviour injunctions and/or +criminal behaviour orders. Although not originally intended, the Matrix is also allegedly used in evidence to +support the prosecution of gang-related offences. Individuals listed on the Matrix have been subjected to police +stop and search more frequently than the general population, with subsequently reduced levels of police +attention for those whose details are removed. Individuals identified as potential ‘victims’ of gang crime and +other vulnerable persons may also be included in the Matrix, albeit accompanied by a ‘zero harm’ score, as are +individuals designated as Gang Nominals with no previous criminal convictions. + +2.2 The Harm Assessment Risk Tool (HART) + +HART is an algorithmic tool that, until recently,8 was used by Durham Constabulary to help custody officers +decide whether arrested individuals should be offered an opportunity to participate in its ‘Checkpoint’ +rehabilitation program. HART was developed collaboratively by Durham Constabulary and Cambridge +University researchers using ‘custody event data’9 drawn from the Constabulary’s custody management IT +systems for the five years to 31 December 2012.10 Considerably more information about the HART tool is +available than is typically the case for public sector decision-support tools, thanks to Sheena Urwin (then Head +of Criminal Justice at Durham Constabulary), who wrote her Master’s thesis under the supervision of Dr + +9 Amnesty International United Kingdom Section (“Amnesty International”), “Trapped in the Matrix: Secrecy, stigma, and +bias +in +the +Met’s +Gangs +Database” +(May +2018) +20 +https://www.amnesty.org.uk/files/reports/Trapped%20in%20the%20Matrix%20Amnesty%20report.pdf, 13. +7 MOPAC, “Review of the MPS Gangs Matrix” at https://www.london.gov.uk/sites/default/files/gangs_matrix_review_- +_final.pdf, 20; Amnesty International, “Trapped in the Matrix,” 11. +8 House of Lords Justice and Home Affairs Committee, “Technology rules? The advent of new technologies in the justice +system” (HL 2021-2022, 180-1). +9 A ‘custody event’ refers to the disposal decision taken by the custody officer following arrest at the end of the first custody +period, either to grant bail, remand D in custody, taken no further action, administer an out of court disposal, or prosecute +the subject (with a decision to bail): Sheena Urwin, “Algorithmic forecasting of offender dangerousness for police custody +officers: An assessment of accuracy for the Durham Constabulary model”, Research Presented as for the purposes of +gaining a Master’s Degree in Applied Criminology and Police Management at Cambridge University (2016) at +http://www.crim.cam.ac.uk/alumni/theses/Sheena%20Urwin%20Thesis%2012-12-2016.pdf, 37. +10 Urwin, “Algorithmic forecasting of offender dangerousness,” 37. See Teresa Scantamburlo, Andrew Charlesworth and +Nello Cristianini, “Machine Decisions and Human Consequences,” in K. Yeung and M. Lodge (eds.), Algorithmic +Regulation (Oxford: Oxford University Press, 2019) which explains the technical development of the HART tool. + + + + + + +4 +Geoffrey Barnes - one of the academics credited with designing HART.11 Urwin’s thesis,12 which she published +online, explains that HART used thirty-four risk predictors based on data about the arrested person at the time +of arrest, combined with data from Durham Constabulary’s pre-existing records, created using a random forest +ML modelling technique. HART purported to calculate the arrestee’s risk of committing an offence in the +subsequent two years, described as a prediction of ‘offender dangerousness’ (a misnomer: those arrested have +not been convicted and were therefore wrongly described as ‘offenders’). Individuals predicted as likely to +commit a ‘serious’ offence (defined as an offence involving violence), a non-serious offence, or no offence, +were classified as ‘high’, ‘moderate’ or ‘low’ risk respectively.13 Only those receiving a ‘moderate’ prediction +were eligible for Checkpoint. Checkpoint participants are provided with an ‘out-of-court disposal’ which +involves signing a 4-month deferred prosecution contract. Provided they meet the contract conditions +throughout (including refraining from offending)14 participants avoid formal prosecution and a criminal record. +Individuals could not be diverted into Checkpoint unless the custody officer was satisfied that there was +sufficient evidence to charge, in line with Crown Prosecution Service requirements. + +2.3 +SyRI + +SyRI was an ML-based decision-support tool used by several Dutch public authorities to help detect tax and +social security fraud until it was declared unlawful following a successful judicial review challenge.15 SyRI was +built from data extracted from individual’s administrative records held by government agencies to generate +individual risk profiles to identify suspected welfare fraud (i.e. those with “an increased risk of irregularities” +when compared to the target population).16 Potential risk indicators were based on demographic data (e.g., name, +address, and associated administrative data), administrative fines and sanctions, debt burdens, social benefit, +and tax. If algorithmically flagged, the individual’s profile was then investigated by the Ministry of Social +Affairs and Employment. Little is publicly known about the substantive interventions that followed once +individuals had been notified that they had been flagged for investigation, but reportedly include administrative +sanctions/fines, mandatory participation in civic education programmes, or in sufficiently serious cases, referral +to the Public Prosecution Service.17 In short, SyRI offered Dutch authorities a powerful technological means for +monitoring individuals in real-time, linking vast swathes of personal, economic and administrative data +collected from records held by multiple governmental organisations to inform how public authorities exercise +their investigative and enforcement discretion. + +3. +Understanding tool-building through the lens of ‘algorithmic regulation’ + +In Part I, we explained how statistical prediction models built using ML techniques conventionally entails +making abstraction decisions that intentionally ignore context-specific features of the real-world domain in + +11 Geoffrey Barnes was then an Affiliated Lecturer at Cambridge University’s Institute of Criminology; see Josh Jacobs +‘The +radical +idea +to +reduce +crime +by +policing +less, +not +more’ +(Wired, +10 +March +2021) +https://www.wired.co.uk/article/evidence-based-policing. Information about HART is available from an academic journal +article jointly written by Barnes, who was responsible for its technical design, and Urwin, who occupied a senior role in +the Durham Constabulary at the time of its use - along with legal scholars Oswald and Grace: Marion Oswald et al., +“Algorithmic risk assessment policing models: lessons from the Durham HART model and ‘Experimental’ proportionality” +(2018) 27(2) Information & Communications Technology Law 223-250. Although the paper claims to offer a ‘critique’ of +the HART tool, we do not understand why the authors declared that there was “no potential conflict of interest” at 250. +12 Urwin’s thesis sought to assess the validated accuracy of the HART model: “Algorithmic forecasting of offender +dangerousness.” 37; Oswald et al, “Algorithmic risk assessment policing models,” 229-230. +13 Urwin, “Algorithmic forecasting of offender dangerousness,” 15. +14 Contracts can include up to five conditions: e.g., refraining from re-offending, participating in restorative processes with +victims, attending mental health or substance abuse therapy sessions, or wearing a GPS tag: Kevin Weir, Gillian Routledge +& Stephanie Kilili, “Checkpoint: An Innovative Programme to Navigate People Away from the Cycle of Reoffending: +Implementation Phase Evaluation” (2019) 15 Policing: A Journal of Policy and Practice 1-19, 9. +15 NJCM et al. and FNV, Rechtbank den Haag, ECLI: NL: RBDHA: 2020:1878 (“SyRI Judgment”). +16 E.g., the Tax and Customs Administration, municipal authorities, the Public Prosecution Service, the police, and the +immigration service: SyRI judgment, [3.7]. +17 SyRI Judgment at [4.17]; [6.59]. + + + + + + +5 +which the tool will operate. For algorithmic tools intended to inform criminal justice decision-making, ignoring +these context-specific features includes ignoring the larger constitutional backdrop against which these +decisions are made. We now demonstrate more precisely how particular design-choices in algorithmic tool- +building implicate, and must be constrained by, public law principles and legal duties by adopting the lens of +‘algorithmic regulation.’18 This analytical approach begins from the perspective of the data scientist +commissioned to build an algorithmic tool for a specific organisational purpose, tracing the steps and design +choices involved and then proceeds to critically investigate the points of contact between the algorithmic tool +and the social world of its deployment, particularly their impacts upon those subjected to algorithmic evaluation. +Although this second stage allows many different analytical perspectives,19 we focus on the constitutional +principles that a commitment to constitutionalism would impose as non-negotiable requirements – or +‘parameters’ to use more familiar computer science terminology – that should condition those choices. By +decomposing ML tool-building into four-steps, we demonstrate in concrete terms how design choices implicate +constitutional principles and the specific legal duties that operate to inform and constrain those choices. + +For simplicity, we consider a specific set of decisions made under conditions of uncertainty involving some +kind of classification task: such as distinguishing ‘spam’ emails from legitimate (‘ham’) emails, to identify +whether a patient is a diabetic or not, or to evaluate whether a person currently serving a custodial sentence for +a criminal offence will re-offend on release. Imagine an organisation whose front-line decision-makers are +routinely required to make decisions of this kind, and commissions the building of an algorithmic tool to assist +them. To create such a tool, the algorithmic model-building process proceeds in four steps20: + +Step 1: Identify suitable input data to serve as ‘ground truth’ for model-building; + +Step 2: Create an algorithmic model by applying ML techniques to generate predictions concerning +future outcomes on unseen data; + +Step 3: Test and validate the model; and + +Step 4: Encode the model into a software program capable of communicating its outputs via a digital +interface to create a convenient digital tool to assist front-line workers. + +3.1 +Step 1: Obtain a suitable dataset to serve as ground-truth to train a prediction model + +First, the developer must identify a dataset to serve as ‘ground truth’ in which each data item possesses the +relevant feature of interest upon which to train the model.21 Properties used to describe the item to be classified +are often called ‘features’ while the classes assigned to each item are called ‘labels’.22 The relevant features of +any given item depend upon the phenomenon which the developer chooses to rely on as the basis for +classification. For our three examples, the features might refer respectively to: words used in the email, a set of +clinical indicators, or a person’s criminal history. The labels applied to the training data comprised of these +items might be respectively labelled ‘spam’ or ‘ham’, ‘diabetic’ or ‘healthy’, or ‘safe’ or ‘dangerous.’ A +machine learning algorithm may then be applied to this labelled training data to build a mathematical function +(a ‘classifier’) that assigns a class label (e.g., ‘spam’ or ‘ham’) to any object (e.g., emails, patients, prisoners) +that has not yet been labelled.23 The resulting classifier can be used to make predictions on unseen data, helping +to inform (or even to automate) decisions for a specific organisational purpose, such as the creation of an + +18 Karen Yeung, “Algorithmic regulation: A critical interrogation” (2018) 12 (4) Regulation and Governance 505-523. +19 See Lena Ulbricht and Karen Yeung “Algorithmic regulation: A maturing concept for investigating regulation of and +through algorithms” (2021) 16(1) Regulation and Governance 3-22 on the ‘thin’ nature of algorithmic regulation as an +analytical frame and the ‘thicker’ variety of conceptual lenses it allows. +20 For a basic explanation of how ML-based models utilising supervised learning are created, see Part I section 1. +21 David Lehr and Paul Ohm, “Playing with the Data: What Legal Scholars Should Learn About Machine Learning” [2017] +51 U.C. Davis L. Rev. 654 – 717, 676; M. Hildebrandt, “Algorithmic regulation and the rule of law” (2018) 376 Phil. +Trans. R. Soc. A 1-11, 7. +22 Lehr and Ohm, “Playing with the Data,” 665. +23 Scantamburlo et al., “Machine Decisions and Human Consequences,” 53. + + + + + + +6 +automated email-filtering tool, a screening tool to help identify patients likely to be (or become) diabetic, or to +assist a parole board deciding whether to grant a prisoner early-release. +This model-building process relies on several critical assumptions. First, that the training data serves as +appropriate ‘ground truth’, a matter to which we will return. Secondly, that historic data provides a reliable +guide to future instances of the phenomenon it is taken to represent: this assumes that the relevant practices and +their surrounding context remain stable and unchanged, and that the training data was collected in a context +relevantly similar to the proposed context of deployment: e.g., new forms of spam might not be accurately +identified by a spam-predictor that has not included training data with that kind of spam. However, the +conventional training of computer scientists might not extend to matters concerning the collection of training +data and/or its relationship to real-world phenomenon which their models seek to predict beyond a narrow, +relatively limited set of technical requirements. That said, the work of data scientists involves attending to the +‘quality’ of datasets to render them useful for algorithmic modelling, although data ‘quality’ is context +dependent. Poor quality data includes ‘dirty’ data, suffering from problems that later hinder the quality of a +given algorithmic model thus requiring human ‘cleaning’4 to render the data suitable for model-building,24 or +the data may be incomplete, noisy, contain significant outliers, infused with errors (caused by a variety of human +and computer fallibilities, such as a failure in code, or a failure in human data entry) or formatted in a manner +that is not processable by modelling software. As the European Fundamental Rights Agency (FRA) has +observed, the use of ‘low quality’ data to produce prediction models that produce ‘low quality’ outcomes to +inform real-world decision-making can lead to violations of fundamental rights, particularly rights to privacy +and data protection25 while negatively affecting the right to an effective remedy, lamenting the failure of many +textbooks and articles dealing with data science to overlook these crucial aspects of data quality.26 Identifying +and selecting the appropriate input and training data requires active involvement and normative judgement by +the data scientist to ‘wrangle’ a dataset into a useable format. However, even if technical developers regard +themselves as professionally responsible for some matters of data quality, this might not extend to the legality +of data collection and processing.27 For algorithmic tools intended to assist criminal justice authorities, routinely +collected administrative data is often used. Yet even assuming that such data can be lawfully collected, it does +not necessarily follow that it can be lawfully used to build algorithmic tools28 for at least two reasons: +(a) Unlawful data collection and processing + +Firstly, privacy laws and contemporary data protection laws restrict the collection and processing of ‘personal +data’ (i.e., data pertaining to an identified or identifiable individual) in significant ways.29 Algorithmic tools +used for criminal justice purposes have been found to violate these laws.30 For example, the Hague District +Court ruled that the state could lawfully create and employ data-driven technologies to identify fraudulent + +24 Tarleton Gillespie, ‘The Relevance of Algorithms’ in T. Gillespie, P.J. Boczkowski and K. Foot (eds.) Media +technologies: Essays on Communication, Materiality, and Society (MIT Press, 2014), 171. +25 European Union Agency for Fundamental Rights, “Data quality and artificial intelligence: mitigating bias and error to +protect fundamental rights” (11 June 2019) https://fra.europa.eu/sites/default/files/fra_uploads/fra-2019-data-quality- +and-ai_en.pdf. +26 European Union Agency for Fundamental Rights, “Data quality and artificial intelligence,” 3. +27 Andrew Selbst, Suresh Venkatasubramanian and I. Elizabeth Jumar, “The legal construction of black boxes: How +machine learning practice informs foreseeability” (2021) Paper presented at the We Robot 2021 conference, +https://werobot2021.com/wp-content/uploads/2021/08/Kumar_et_al_Legal-Construction-of-Black-Boxes.pdf. (cited with +the kind permission of the authors). +28 It might be lawful to use some data train a model, but not to use the same set of features about a person to make/inform +a decision about them using the model in the real world, or vice-versa. See ICO guidance on why controllers need distinct +lawful bases for development and deployment: Information Commissioner’s Office, “What do we need to do to ensure +lawfulness, fairness, and transparency in AI systems?” at https://ico.org.uk/for-organisations/guide-to-data- +protection/key-dp-themes/guidance-on-ai-and-data-protection/what-do-we-need-to-do-to-ensure-lawfulness-fairness- +and-transparency-in-ai-systems/?q=profiling. We are grateful to Reuben Binns for drawing this to our attention. +29 Article 4(1) of the EU’s General Data Protection Regulation 2018 provides that ‘personal data’ applies only to +information ‘relating to an identified or identifiable natural person (‘data subject’).’ See Case C-582/14 Breyer v Germany +[2014] ECLI:EU:C:2016:779). +30 Although there is room for debate about whether the resulting model could be considered unlawful per se. + + + + + + +7 +activity, SyRI was disproportionately intrusive and thus not legally justified as ‘necessary in a democratic +society’, violating the Art 8(1) right to private life.31 In a related vein, the UK Information Commissioner’s +Office (ICO) ruled that by naming identifiable individuals and labelling them ‘Gang Nominals’, the Gangs +Matrix violated data protection laws. Although the data parsed by the Matrix to generate individual harm scores +is drawn primarily from administrative data lawfully collected by the London MPS, it relied upon the excessive +collection of personal data including ‘gang association data’ on informal unregulated lists, entailed unregulated +data sharing between partner agencies, and included victim data in the Matrix without distinguishing them from +suspected ‘Gang Nominals.’32 +(b) Increasing the likelihood of the unlawful exercise of public decision-making authority + +Secondly, developers may regard the legal duties applicable to public authority decision-making as matters as +outside their problem-space. But without a proper understanding of these duties, the tools they build may +facilitate unlawful decision-making. Moreover, decisions attached to a particular office (including those taken +by criminal justice officials, such as the police) are typically accompanied by specific legal duties, including +administrative law duties that restrict the lawful exercise of discretionary power in the hands of public officials. +For example, British administrative law requires that public power be exercised in accordance with the purpose +for which that power was conferred, and only on the basis of ‘relevant considerations,’ while ‘irrelevant’ +considerations must be excluded.33 Accordingly, data used to train the algorithmic model must generate outputs +that are legally ‘relevant’ to the decisions those outputs inform. But if the phenomena that the training data is +taken to represent does not conform with legal requirements, then the resulting model may generate predictions +that are legally ‘irrelevant’ to the decision at hand.34 This might occur, for example, if the training data used is +not a valid or lawful indicator of the predicted phenomenon. So, if legislation requires consider of variable A +only, it is not legally permissible for the decision-maker to consider variable B.35 Yet if no ‘ground truth’ data +concerning the matter which a public authority is legally required to consider when making particular decisions +is available, data scientists might utilise data-sets they regard as a ‘proxy’.36 This creates serious risks that the +model’s outputs will be legally irrelevant to the decision at hand resulting in decision-making that is ultra vires +and therefore unlawful. +For example, consider a police officer in England or Wales who is legally required to decide whether an arrested +person in custody (D) should be released without charge (or under continuing investigation), charged and +publicly released on bail, or charged and detained in custody on remand prior to trial.37 Continued detention on +remand pending trial is only lawful if the conditions of s 3(6) of the Bail Act have been met and authorised. +This requires establishing (among other things) that detention is deemed ‘necessary’ to secure D from +absconding, to prevent D committing an offence while on bail, or to protect D or another person.38 It is in making + +31 ECHR Art.8. +32 Information Commissioner’s Office, “Enforcement Notice to the Commissioner of Police of the Metropolis” (13 +November 2018), https://www.met.police.uk/SysSiteAssets/media/downloads/force-content/met/about-us/gangs-violence- +matrix/ico-enforcement-notice.pdf. +33 R v Secretary of State for the Home Department, ex parte Venables and Thompson [1998] AC 407 +34 Marion Oswald, ‘Algorithm-assisted decision-making in the public sector: Framing the issues using administrative law +rules governing discretionary power’ [2018] 376 Phil. Trans. R. Soc. A 1-20, 10; +35 The matter may be arguable if variable A can be consistently and reliably inferred from variable B. Difficulties can arise +in evaluating the lawfulness of algorithmic models that have been generated through the use of datasets from which +sensitive attributes (for example, such as gender or race) have been removed, because such sensitive attributes may well +be readily inferred from other features in the data: see A. Roth and M. Kearns, The Ethical Algorithm: The Science of +Socially Aware Algorithm Design (Oxford: OUP, 2019), 74-84. +36 See discussion of the limitations of proxy data within the contexts of education, social care and criminal justice in Abigail +Z. Jacobs and Hanna Wallach, “Measurement and Fairness” (2021) FAccT’ 21 Proceedings of the 2021 ACM Conference +on Fairness, Accountability and Transparency, 375-385. +37 For a more expansive illustration of this decision process in the context of the use of algorithmic tools, along with +applicable laws, see section 3.3. of Part I. +38 Bail Act 1976, s.3(6): the decision whether to refuse bail and to remand D pending trial is made by a magistrate. + + + + + + +8 +these kinds of determinations that US criminal justice authorities have enthusiastically embraced algorithmic +tools that purport to assess the risk that an individual will abscond (‘flight risk’) or commit a criminal offence +if released (so-called ‘recidivism risk’),39 although we are not aware of their use by British courts for these +purposes. In assessing the necessity of retaining D in custody for the purposes permitted by the Act, some kind +of assessment of the likelihood that the individual will commit a criminal offence, engage in self-harm, or harm +another person during the period prior to trial is needed. To create an algorithmic model capable of providing +such predictions, ‘ground truth’ data would consist of a comprehensive historic data set identifying the entire +universe of crimes, and of individuals who have engaged in criminal conduct or otherwise harmful conduct +directed at others or themselves for the geographic region in question over a substantial yet recent time-period. +Yet because not all crimes or harmful actions are identified, reported, and recorded, and many of those who +commit crimes and/or harms evade apprehension and conviction, in addition to a lack of systematic data +concerning self-harming activities falling short of suicide, ground truth data about the commission of crime and +harm across a population, even within a more narrowly defined geographic area, is not available and, indeed, is +likely impossible to collect. This has not, however, prevented developers from building so-called ‘recidivism +risk’ predictors using arrest data, treating arrest as a ‘proxy’ for crime committed. Arrest data is, however, an +inadequate, highly misleading indicator of crime because not all arrested persons are charged, those who are +charged may not be convicted, and many crimes are committed for which no arrests are made. Accordingly, the +mere fact of arrest is not an acceptable proxy for the commission of criminal offence. +Thus, by failing to consider what the training data actually signifies, the tool’s outputs may not offer meaningful +indications of the phenomenon it purportedly predicts. Accordingly, the tool’s outputs may be ‘legally +irrelevant’ to the matter the public official must decide and therefore cannot lawfully be considered. In the above +example, it would not be legally permissible to remand in custody an arrestee charged with a criminal offence +solely on the basis of an algorithmic tool that predicts the likelihood that D will be re-arrested because the legal +significance of a lawful arrest differs very significantly from that of a criminal conviction.40 But it is on the basis +of arrest data, for example, that the HART tool was developed, which is described as generating predictions +about whether an individual is likely to ‘commit a criminal offence’ within a two year period. At the very least, +these tools should be properly described as ‘arrest predictors’ rather than mislabelling and misrepresenting them +as predictions of future criminal offending.41 +3.2 +Step 2: Apply a machine learning algorithm to create a prediction model + +Armed with labelled ‘ground truth’ data, ML software is then applied to train and build an algorithmic model +(a ‘classifier’) that assigns a class label (e.g., ‘spam or ham’, ‘diabetic or healthy’, ‘safe or dangerous’) to unseen +data. Advanced ML techniques enable the generation of more powerful and sophisticated mathematical models +by analysing larger, more complex, highly dimensional data. Because many ML methods are available, +technical developers must choose between them (and may experiment with several alternatives) including +regression models that generate a score for each candidate or classification models that allocate candidates into +classes. One parameter of great importance, particularly for algorithmic tools used in rights-critical contexts, +concerns the configuration of the model’s ‘error thresholds.’42 Predictions generated using ML models are +inherently probabilistic: prediction errors are therefore unavoidable. In this context, errors are defined as the +failure of the model to generate an accurate prediction on unseen data, either in the form of a Type I error (false +positive) or Type II (false negative).43 Good data scientists also recognise the vital importance of attending to +the distribution of Type I and Type II errors in building algorithmic models, requiring careful consideration of + +39 Kathrin Hartmann & Georg Wenzelburger, “Uncertainty, risk and the use of algorithms in policy decisions: a case study +on criminal justice in the USA” (2021) 54 Policy Sciences 269-287. +40 Virginia Eubanks, Automating Inequality: How High-Tech Tools Profile, Police and Punish the Poor (New York: St. +Martin’s Press, 2018). David Robinson “The Challenges of Prediction: Lessons from Criminal Justice.” (2017) 14 Journal +of Law & Policy for the Information Society Robinson, 151. +41 Robinson, “The Challenges of Prediction,” 160-161. +42 Steven W. Knox, Machine Learning: A concise introduction (Chichester: Wiley, 2018), 15-32; D. Spiegelhalter, The Art +of Statistics: Learning from Data (Pelican, 2020). +43 Spiegelhalter, The Art of Statistics: Learning from Data. + + + + + + +9 +their real-world consequences. For example, consider the consequences of errors produced by an algorithmic +tool that evaluates skin melanoma images to predict whether a melanoma is cancerous or benign, intended for +at-home diagnosis to help individuals decide whether to seek medical advice. A Type II error (false negative) +is considerably more serious: a person who mistakenly believes that the melanoma is benign may not seek +further medical treatment, allowing the cancer to develop untreated, with potentially fatal consequences. In +contrast, a Type I error (false positive) is likely to prompt that person to seek medical assistance unnecessarily, +assuming that the melanoma is subsequently diagnosed correctly by the clinician as benign.44 + +Careful consideration of the real-world consequences of different error-types is particularly important when +determining the cut-off point at which point the model assigns a classification. As Scantamburlo et al explain, +when ML techniques are used to build binary classifiers that can assign an item to one of many possible +categories (such as ‘spam’ or ‘ham), many (although not all) involve in a two-stage process: first, a real valued +score is computed for the item to be classified, and secondly, that score is compared with a cut-off point +whereupon the item is assigned to a class depending on whether it exceeds the cut-off point.45 The real valued +score could informally be thought of as a probability, though it is not necessarily a formal probability. In some +(but not all) cases, changing the threshold results in a trade-off between false positives (Type I error) and false +negatives (Type II error). In the example of skin-cancer predictor, this can be captured by notions of ‘sensitivity +or recall’ (i.e., the true positive rate is the proportion of true positives in relation to the aggregate of true positives ++ false negatives) and ‘specificity’ (i.e., the true negative rate is the proportion of true negatives in relation to +the aggregate of true negatives + false positives). For example, for our melanoma classifier, because the +consequences of mistakenly classifying a melanoma as benign when it is in fact cancerous (Type II error) is far +more serious than mistakenly classifying a benign melanoma as cancerous (Type I error), the cut-off point +should be configured to minimise Type II errors, even at the cost of increasing Type I errors. These examples +also demonstrate that the real-world consequences of error are context-dependant: a Type I error in ‘recidivism +risk’ prediction is far more serious for an affected individual than a Type I error by an automated spam-filter. + +As a matter of constitutional principle, algorithmic tools that inform how an individual will be treated by +criminal justice authorities must be configured to distribute the risk of error in a manner that respects due process +rights, including the presumption of innocence protected under ECHR Art 6. These procedural rights are rooted +in the state’s duty to demonstrate respect for persons. It demands that those adversely affected by the decision +of a public authority, particularly criminal justice authorities wielding the coercive power of the state, are +entitled to be informed of, and to challenge those decisions, particularly given the ever-present danger of +mistakes in decision-making. As a matter of administrative law, the specific procedural requirements (and +concomitant duties) that the right to due process demands is highly context-dependent, in which the decision’s +impact upon the affected individual’s rights, interests and legitimate expectations has great importance.46 Hence +a person whose driver’s licence application is refused enjoys a fairly limited set of procedural rights,47 while a +person charged with committing a very serious criminal offence is entitled to an extensive suite of procedural +rights, typically including rights to free legal representation, to the application of strict rules of evidence and +procedure, and to the presumption of innocence in which the burden of providing the accused’s guilt ‘beyond +reasonable doubt’ is placed firmly on the prosecution.48 This high standard of proof, which is far more +demanding than the ‘balance of probabilities’ standard applicable to civil cases, produces a legal system +systematically designed to minimise Type I errors (false positives) while producing more Type II errors (false +negatives). This means guilty defendants may avoid conviction because the prosecution has failed to prove guilt +‘beyond reasonable doubt.’ Although such outcomes are deeply regrettable, these errors are nevertheless widely +accepted in modern western European legal systems as worth incurring to avoid the far more egregious injustice +associated with Type I errors—that is, wrongfully convicting an innocent person. Particular care is needed for + +44 I.e., the patient will not then be subjected as very invasive, unnecessary interventions due to an incorrect diagnosis. See +for example Karsten Juhl Jørgensen, Peter C. Gøtzsche, Mette Kalager, and Per-Henrik Zahl. “Breast Cancer Screening in +Denmark: A Cohort Study of Tumor Size and Overdiagnosis” [2017] 166(5) Ann Intern. Med. 313-323. +45 Scantamburlo et al., “Machine Decisions and Human Consequences,” 54. +46 D. Galligan, Due Process and Fair Procedures: A Study of Administrative Procedures (Oxford: OUP, 1996). +47 Including rights to be tested by an unbiased examiner, to be given a reasonable opportunity to demonstrate that their +driving skills meet the requisite legal standard, and to be provided with reasons for any refusal of a license. +48 ECHR, Art. 6. + + + + + + +10 +prediction models created by ML techniques which exacerbate the dangers associated with mistakes because +they are produced on the basis of correlations in the underlying training data rather than on scientifically +established causal relationships.49 + +To demonstrate how human rights should condition algorithmic tool-design, consider again the decision to +detain or release a person charged with a criminal offence (D) pending trial. To detain a person against her will +for extended periods entails a grave and serious denial of liberty. If these decisions are to be informed by +predictions that purport to classify an individual as ‘dangerous’ or ‘harmless’ such that the former is held in +remand while the latter released, then falsely classifying a harmless person as ‘dangerous’ (Type I error) will +entail a serious interference with that person’s right to liberty under Article 5 ECHR.50 However, falsely +classifying as ‘harmless’ a person (‘D’), who is in fact dangerous, will result in release, placing the public at +risk that D will commit a serious crime during the pre-trial period. Nevertheless, if we are to take the +presumption of innocence and the individual’s right to liberty seriously, this necessitates treating false positives +(convicting the innocent) as far more serious than false negatives (failing to convict the guilty). As a matter of +constitutional principle, this relative weighting must be reflected in the construction of the underlying +mathematical model, although the human decision-maker might justifiably override the algorithmic +recommendation if release may expose specific, vulnerable individuals to a substantial risk of domestic +violence. Although this means the public systematically bears a proportionately greater risk that a dangerous +person will be released pending trial, constitutional democratic societies accept that this is the inescapable cost +of upholding individual rights and freedoms. If the model were to be configured otherwise, this may unlawfully +violate the arrestee’s Article 5 ECHR right to liberty and security, including freedom from arbitrary detention, +and will also increase the likelihood of further violation of her Article 6 ECHR rights, including the right to be +presumed innocent and the right to a fair trial. + +Despite the importance of these technical choices, there is little public information concerning how the risks of +Type I and II error are configured into algorithmic tools used by criminal justice authorities. Of our three case +studies, public information was only available in relation to the configuration of error thresholds for the HART +tool, thanks to Urwin’s published Master’s thesis and the account provided in a jointly authored paper with +Barnes, Oswald and Grace.51 Yet those accounts reveal a disturbing failure to demonstrate any recognition that +an arrested person is entitled to be presumed innocent, stating that error thresholds were configured on the basis +that “it is worse to misclassify a dangerous offender as harmless than to erroneously classify a harmless +individual as dangerous.”52 Although this may reflect the perspective favoured by the general public, it fails to +recognise that such a choice entails an unjustified violation of the affected individual’s right to be presumed +innocent (and hence harmless).53 + +3.3 +Step 3: Model testing and validation + +Having generated a mathematical model, computer scientists then conventionally assess its ‘accuracy,’ +understood as how well it correctly predicts outcomes from a set of unseen historic data. Binary classification + +49 See section 3.1 of Part I for further discussion of the constitutional acceptability of using ML models to make predictions +about individuals in rights-critical contexts. +50 ECHR, Art. 5. +51 Oswald et al, in “Algorithmic risk assessment policing models,” state at 236: ‘The HART model represents a real example +of a value-judgement built into an algorithm, so requiring a ‘trade-off’ to be made between false positives and false +negatives in order to avoid errors that are thought to be the most dangerous: in this context, offenders who are predicted to +be relatively safe, but then go on to commit a serious violent offence (high risk false negatives). As a consequence, high +risk false positives have been deliberately made more likely to result.’ +52 Urwin, “Algorithmic forecasting of offender dangerousness.” +53 Further, HART’s error thresholds were unjustified because they were insufficiently tailored to the policy purpose for +which it was introduced to serve. Neither high risk false positives, where the individual was actually low risk, nor high +false negatives where the individual was actually high risk, result in any change in the substantive recommendation. High +and low risk scores could not grant entry into Checkpoint. Only if a person who is medium risk is wrongly classified as +high or low risk would they be erroneously denied entry. If individuals who are low or high risk were erroneously classed +as medium risk, they would then be granted entry, when in fact the policy intention was to exclude them. On the +presumption of harmlessness, see A. Ashworth and L. Zedner, Preventive Justice (Oxford: OUP, 2014), 53. + + + + + + +11 +algorithms of the kind under consideration are conventionally and primarily evaluated in terms of accuracy in +predicting the correct classification of candidates taken from the ‘test set’ (or ‘out of bag sample’) namely data +held back and kept separate from the remainder of the dataset used to train and tune the model.54 Accuracy in +this context conventionally refers to the percentage of accurate predictions calculated as a ratio of the total +number of correct predictions (that is, the total number of true positives and false negatives) relative to the total +number of predictions generated by the model.55 But there are many other commonly used quality metrics that +can be employed to evaluate a tool’s ‘accuracy’ with Krafft and Zweig56 identifying 31 different and ‘frequently +used’ quality measures. These include, for example, precision and recall, which we have already considered57, +as well as the AUC-ROC curve.58 They argue that developers should select quality measures that are most suited +to the social and organisational context in which the algorithm is to be used, yet they observe that this is not +always the case.59 Furthermore, these mathematical assessments of ‘accuracy’ typically overlook the fact that +the datasets themselves are of dubious validity as a basis for predicting what an individual will do if publicly +released: although they will include data on whether a person who was released was subsequently arrested, they +will not contain data about whether those detained in custody would have committed a crime had they been +released. + +There is, however, increasing recognition by data scientists of the need to evaluate ‘quality’ from values other +than accuracy in prediction. Those working within the field of fair-ML or ‘FAccT’ computing are actively +investigating how ML models may be evaluated by reference to values such as ‘fairness,’ ‘explainability’ and +‘transparency.’60 Yet these approaches typically adopt a very narrow and somewhat contrived understanding of +normative values, conceived largely in mathematical terms which then lend themselves to quantification and +computational analysis.61 This is particularly true of ‘fairness’, with researchers seeking to define aspects of +inherently vague notions of societal fairness in mathematical terms in order to incorporate fairness ideals into +machine learning.62 However, as critics observe, these definitions are simplifications that fail to capture the full +range of similar and overlapping notions of fairness and discrimination in philosophical, legal and sociological +contexts.63 + +Apart from the dangers associated with relying upon ML-based prediction models for which a causal +understanding of the variables remains unknown (examined in Part I), these quality metrics focus exclusively +on the algorithmic tool rather than the larger socio-technical system in which they operate producing a ‘framing + +54 D. Michie, D. Spiegelhalter and C. Taylor (eds.) Machine Learning, Neural and Statistical Classification (New Jersey: +Prentice Hall, 1995), 15. A ‘validation set’ refers to at least one portion of data from the training dataset conventionally set +aside to validate the model and tune its parameters. +55 Kaitlin Kirasich, Trace Smith and Bivin Sadler, “Random forest vs logistic regression: Binary classification for +heterogenous datasets” (2018) 1(3) SMU Data Science Review 1-24, 12-14. +56 Tobias D Krafft and Katharina A Zweig “So far, So Good: Multidisciplinary perspectives on algorithms, decisions and +algorithmic decision-making – Computer Science Dimensions.” (2019) Unpublished manuscript. +57 See section 3.2. +58 Area under the curve – receiver operating characteristic: Melissa Hamilton “Adventures in risk: predicting violent and +sexual recidivism in sentencing law” [2015] Arizona State Law Journal 47(1) 11-62, 34-37. +59 E.g., demonstrating that the purported AUC-ROC score of COMPAS (an algorithmic tool developed by Northpointe Inc. +to undertake ‘recidivism risk’ assessment and inform sentencing decisions), offers very little meaningful information about +the tool’s accuracy in, nor suitability for, this purpose: Krafft and Zweig ‘So far, So Good…’ at 8. +60 +e.g., +see +the +ACM +Conference +on +Fairness, +Accountability, +and +Transparency +(ACM +FAccT): +https://facctconference.org/; Roth and Kearns, The Ethical Algorithm: The Science of Socially Aware Algorithm Design. +61 C. Barabas, “Beyond bias: ‘Ethical AI’ in criminal law” in M. Dubber, F. Pasquale and S. Das (eds.), The Oxford +Handbook of Ethics of AI (OUP, 2020). +62 See the well-known debate about the fairness of COMPAS risk scoring: Julia Angwin, Jeff Larson, Surya Mattu, and +Lauren Kirchner, 'Machine bias: There’s software used across the country to predict future criminals. And it’s biased +against blacks” (ProPublica, 2016) at https://www.propublica.org/article/machine-bias-risk-assessments-in-criminal- +sentencing; William Dietrich, Christina Mendoza and Tim Brennan, “COMPAS risk scales: Demonstrating accuracy equity +and +predictive +parity +(2016)” +Northpointe +Inc. +https://go.volarisgroup.com/rs/430-MBX- +989/images/ProPublica_Commentary_Final_070616.pdf. +63 Ben Green, “The false promise of risk assessments: Epistemic reform and the limits of fairness” (2020) FAT*’20: +Proceedings of the 2020 Conference on Fairness, Accountability and Transparency, 594-606; Roth and M. Kearns, The +Ethical Algorithm: The Science of Socially Aware Algorithm Design, Ch.2. + + + + + + +12 +trap’ that leads to inappropriate and misleading ‘quality’ guarantees.64 We argue that algorithmic tool ‘quality’ +also demands adherence to the basic constitutional principles and legal requirements, particularly when used to +inform ‘rights-critical’ decisions by public authorities constituting constitute vitally important and relevant +socio-technical context. Hence the ‘quality’ of these tools should be assessed as constitutionally and legally +acceptable before they can be used to inform real-world decisions. These assessments require careful, context- +sensitive, qualitative legal evaluation that cannot be collapsed or reduced to purely mathematical terms. To do +so risks overlooking rights-critical normative choices made by developers in the design process of algorithmic +tools, enhancing the dangers of illegality and injustice. + +3.4 +Step 4: Develop a digital interface for use in a particular organisational setting + +Armed with a suitable prediction model, a digital user-interface is then designed to communicate the model’s +predictions to human decision-makers. The widespread take-up of smart devices substantially enhances the ease +and convenience with which front-line officers can access algorithmic predictions. Although convenience, +efficiency and seamlessness are understandably important from a user-design perspective, there is a worrying +tendency to downplay or overlook the need to design interfaces in ways that respect the due process rights of +affected persons. We have already argued that respect for due process rights should be demonstrated when +configuring error thresholds within algorithmic models. This is also true when designing digital user-interfaces +for front-line decision-makers. Although identifying the specific set of procedural requirements (and +concomitant duties) that the right to due process imposes requires is only possible in specific application +contexts, three matters are especially important yet often overlooked by UX designers trained to prioritise +organisational convenience and efficiency, briefly outlined below: + +i. The right to reasons: under British Administrative law, an individual’s right to reasons for public authority +decisions that have significant adverse effects flows from the legal duty of public authorities to explain and +justify their decisions in accordance with law.65 If an algorithmic tool produces a score and/or recommendation +about an individual without an accompanying explanation of what it is intended to signify, or how it was +generated, the front-line official receiving it may not be able to provide the affected individual with reasons to +support it. Accordingly, user-interfaces for algorithmic tools to support public authority decision-making +should, as a matter of constitutional best practice, if not of legal obligation, provide functional explanations to +accompany the tool’s output(s) (at least for ‘interpretable’ ML models). Front-line officers should also be able +to identify and cross-check how the output was produced, lest over-simplified explanations exacerbate the +likelihood of automation bias, discussed below.66 Similarly, tool outputs must not be mislabelled: a tool intended +to predict ‘recidivism risk’ developed from arrest data should not be called a ‘recidivism’ predictor, but a ‘re- +arrest’ predictor. The legal duty of public authorities to give reasons for decisions that entail the adverse +treatment of individuals suggests that they should not employ decision-support tools that use advanced ML +techniques such as deep neural networks, for which even functional explanations remain elusive.67 Yet, for the +three case studies examined here, we could not identify whether front-line decision-makers were automatically +provided with explanations to accompany the algorithmic predictions intended to assist them. Much more +extensive and systematic transparency regarding decision-support tools used in criminal justice is therefore +needed.68 + +ii. The right to contest: providing explanations about the tool’s functional logic, including the relevant +variables upon which it relies, might help decision-makers discharge their administrative law duty to provide +affected individuals with ‘reasons’ for a particular decision in specific cases. However, those reasons must be +lawful reasons69 and good faith decision-making is not sufficient. Hence the individual’s right to challenge and + +64 Selbst et al., “The legal construction of black boxes.” +65 R v Secretary of State for the Home Department, Ex parte Doody [1993] UKHL 8; Jennifer Cobbe, “Administrative Law +and the Machines of Government: Judicial Review of Automated Public-Sector Decision-Making” 39(4) Legal Studies 63. +66 We are indebted to Mireille Hildebrandt for this point. +67 See Rebecca Williams, “Rethinking administrative law for algorithmic decision-making” (2021) 42(2) Oxford Journal +of Legal Studies 468-494, 482. +68 See section 5. +69 Williams “Rethinking administrative law for algorithmic decision-making,” 482. + + + + + + +13 +contest such decisions has considerable constitutional importance, reflected in rights to liberty, to due process +and to a fair trial protected under ECHR Articles 5 and 6. While the Data Protection Act 2018 (and the GDPR +and Law Enforcement Directive upon which the 2018 Act builds) confers on data subjects a right to contest +fully automated decisions,70 contestation rights must also accrue to those significantly adversely affected by a +recommendation produced by an algorithmic tool to guard against injustice and the abuse of power. + +iii. The right to an unbiased tribunal: Although we have emphasised the ‘fair hearing’ limb of the +administrative law right to procedural fairness, its second limb - the ‘rule against bias’ - is equally important. It +requires that decision-making tribunals must be ‘impartial’, meaning free from both actual bias and the +appearance of bias.71 The issue of bias and discrimination, particularly racial and gender bias of algorithmic +tools used in criminal justice decision-support, has received widespread attention.72 Datasets inevitably reflect +underlying biases in the historic social practices to which the data pertains so that, if used to generate prediction +models, the resulting outputs will reflect and reinforce these biases. Historically marginalised groups are thus +subjected to a higher risk of unjust discrimination relative to individuals from majority groups because the +resulting outputs systematically discriminate unjustly (rather than being arbitrary, they are systematically +patterned for reasons we can point to) and may violate the right to be free from unjust discrimination in the +determination of opportunities and burdens. To this end, the ICO concluded that the London Gangs Matrix was +unlawful because the MPS failed to ensure that its use complied with the Public Sector Equality Duty (s.149 of +the Equality Act 2010) because it disproportionately singles out black men relative to other ethnic groups.73 + +Less attention, however, has been paid to automation bias,74 referring to the human tendency to trust machine- +made judgments over their own despite their potential or demonstrated capacity for error. Although algorithmic +‘recommender’ tools may formally preserve human judgement, front-line officials may in practice tend to +follow recommendations unreflectively.75 As Johnson and Powers76 commented in the context of computerised +aviation systems in which a human ‘in-the-loop’ is expected to supervise computational systems, those +individuals may be understandably reluctant to intervene.77 Similarly, front-line criminal justice decision- +makers under considerable time pressures, heavy workloads and lack a clear understanding of how algorithmic +recommendations are produced are unlikely to depart from them. Accordingly, officials who use algorithmic +tools must be properly trained so that they can properly understand and interpret algorithmic recommendations +and their limitations, and to ensure they exercise meaningful independent judgement rather than unthinkingly +following the tool’s outputs. +4. +Discussion and Recommendations for Reform + +We have demonstrated how important constitutional principles and legal duties are implicated at every step of +the algorithmic model-building process for use by criminal justice decision-makers yet largely overlooked in +relation to three such tools used to date. As a result, these tools may significantly enhance the risk that decisions +based on their predictions may be unjust or may otherwise entail the unlawful exercise of decision-making +authority. To minimise these dangers, it is vital that during the algorithmic tool-building process, proper +consideration is given to: + +(a) the nature of the substantive interventions that flow the outputs generated by these prediction tools, +particularly effects on the rights, interests, and legitimate expectations of those subjected to algorithmic +evaluation; + +70 Williams “Rethinking administrative law for algorithmic decision-making ,” 474-476. +71 Ridge v Baldwin [1964] AC 40. +72 Cathy O’Neil, Weapons of Math Destruction (New York: Crown Books, 2016). +73 Information Commissioner’s Office, “Enforcement Notice to the Commissioner of Police of the Metropolis” at [41]. +74 Linda J. Skitka, Kathleen L. Mosier, Mark Burdick, and Bonnie Rosenblatt, “Automation bias and errors: Are crews +better than individuals?” (2000) 10(1) The International Journal of Aviation Psychology 85-97 (2000); ICO (n 101) at [41]. +75 Hartmann & Wenzelburger, “Uncertainty, risk and the use of algorithms in policy decisions,” 269-287. +76 Deborah G. Johnson and Thomas M. Powers, “Computer systems and responsibility: A normative look at technological +complexity” (2005) 7 Ethics and Information Technology 99-107, 106. +77 Arthur Kuflik, “Computers in control: Rational transfer of authority or irresponsible abdication of autonomy?” (1999) 1 +Ethics and Information Technology 173-184. + + + + + + +14 + +(b) the legal duties and obligations that apply to all public officeholders who make decisions about the treatment +of individuals within the criminal justice process; + +(c) the public policy objectives that the larger criminal justice system in which the tool is embedded is intended +to serve, and + +(d) the need to maintain the general public’s trust and confidence in the integrity of the criminal justice system +and the administration of justice. + +Yet, as we argued in Part I, computer scientists are conventionally trained to abstract or ‘detach’ the prediction +model from legally and constitutionally relevant considerations. By focusing exclusively on developing +mathematical models to generate ‘accurate’ predictions understood in narrow, technical terms, the resulting tool +may violate legal requirements and constitutional principles, resulting in decisions that entail the unlawful or +improper exercise of public power. Despite the intuitive, common-sense need to consider real-world +consequences and legal concerns when approaching the task of algorithmic tool-creation, they tend to be ignored +by technical experts who may conventionally regard them as outside the relevant ‘problem space’ and hence +‘not their responsibility.’ It is therefore essential that technical experts work collaboratively with legal +professionals with a strong grasp of the appropriate constitutional principles, human rights norms and doctrines +of administrative law when creating, testing, and evaluating algorithmic tools prior to their deployment by +public decision-makers.78 However, because these interdisciplinary teams may want more tractable, concrete +guidance to better understand how constitutional principles should apply to, and be ‘operationalised’, in the +tool-building process, we offer brief advice, focusing on three sets of ‘detachment’ practices that should be +steadfastly avoided. + +4.1. +Detaching the tool from the substantive interventions that its predictions seek to inform + +We have shown that algorithmic tools cannot legitimately be designed, evaluated nor deployed without due +consideration of their real-world consequences, requiring careful attention to the specific context of their +deployment and application domain. Yet technical specialists conventionally focus on the accuracy of +algorithmic predictions, without taking due account of the effects of the substantive interventions which follow. +Thus, the consequences of mistaken outputs upon individuals subjected to algorithmic evaluation are not given +due consideration. Although mistaken on-line consumer product recommendations may be inconsequential to +the recipient, this is not true of mistaken, biased, or spurious algorithmic predictions that inform rights-critical +criminal justice decisions, particularly about coercive detention. Throughout the tool-building process, attention +must be paid to the substantive interventions intended to follow from predictions thereby generated, particularly +their resulting impact on the affected individual, to ensure that: + +§ +The model’s error thresholds appropriately reflect the due process rights of the affected individual, +including the right to be presumed innocence (and the presumption of harmlessness) where appropriate; + +§ +Due consideration is given to the normative acceptability of utilising the algorithmic models to generate +predictions to inform decisions that may adversely affect the individual under evaluation, particularly in the +absence of scientific evidence of causal relations upon which the tool relies. Algorithmic tools should not +be deployed in rights-critical contexts unless a plausible account of why the features upon which the model +relies to generate productions may be expected to provide reliable and truthful indicators of the phenomenon +which the model seeks to predict79; and + +§ +The design of the algorithmic tool’s user-interface for communicating outputs to front-line officers must +enable them to understand what the output signifies—including its limitations—and remind the officeholder + +78 Michael Veale, Max Van Kleek and Reuben Binns, ‘Fairness and accountability design needs for algorithmic support in +high-stakes public sector decision-making’ [2018] Proceedings of the 2018 CHI Conference on Human Factors in +Computing Systems 440. +79 For further discussion, see section 3.1. of Part I. + + + + + + +15 +of the rights of affected persons to demand reasons for any significant adverse decision, to contest the +decision, to an unbiased tribunal, and to help the officer ensure that meaningful, independent judgement is +brought to bear when making the resulting decision. + +4.2 +Detaching the tool from its surrounding legal context and/or underlying policy purpose + +Failing to properly attend to the surrounding legal and policy context during the design and construction of +algorithmic decision-support tools enhances the likelihood that these tools will produce predictions that give +rise to unlawful decision-making and may generate serious injustice. Accordingly, when building such tools, +due care must be given to: + +§ +particular matters which the law requires the decision-maker to decide, including specific considerations +mandated by law that must be considered in decision-making. For example, if the statute requires +consideration of the likelihood of an individual committing a crime in future, then it should not be assumed +that this is equivalent to the likelihood of being arrested in future; + +§ +ensuring that only ‘legally relevant’ considerations (‘features’) are analysed in the model generation +process, and that irrelevant features are disregarded; + +§ +attending to the congruence between the phenomenon that the tool is intended to predict and the substantive +decisions which the public official is legally obliged to make. Thus, if the decision-maker is required to +consider the ‘dangerousness’ of an individual, predictions about the likelihood of being arrested in future +may not be reliable indicia of dangerousness; and + +§ +careful evaluation of whether the proposed ground truth data constitutes a valid and acceptable proxy for +generating predictions about the phenomenon of interest. +4.3 +The need for systematic safeguards: attending to the power, opacity, and scale of algorithmic tools + +We have highlighted how algorithmic risk assessment tools used for criminal justice purposes may result in the +unjust and/or unlawful treatment of individuals. But transparency and accountability must also be provided to +the public at large, otherwise there is no basis for them to trust that governmental power is not being abused, +exploited, or otherwise exercised corruptly. Accordingly, the exercise of governmental authority—particularly +in criminal justice contexts—must be subject to systematic, institutional oversight, to ensure that this authority +is exercised in a lawful and accountable manner, including mechanisms to ensure that the exercise of that +authority in individual cases is subject to meaningful contestation, review, and redress. A commitment to +constitutionalism demands that endowed with governmental power exercise it in an open and transparent +manner, routinely rendering an account for the exercise of their decision-making power to the community at +large from whom their power is ultimately derived and on whose behalf they purport to act. Yet relatively little +attention is given to the opacity and power of these tools arising from their scale and speed of operation. Instead, +we observe a third, equally if not more troubling form of ‘detachment’, generating serious constitutional +dangers: the inapposite use of analogical reasoning evident in some policy-making and judicial reasoning, +whereby advanced data-driven tools are detached from their capacity to be employed automatically and at scale. +Instead, algorithmic tools are often treated as ‘equivalent’ to existing (analogue) tools long used to inform +criminal justice decisions, on the basis that they pursue the same purposes. This ‘fallacy of equivalence’ is used +in at least two ways in legal analysis that are both mistaken and dangerous.80 + +Firstly, it is constitutionally mistaken to assume that the legal basis authorising the use of an ‘old fashioned,’ +handcrafted, statistical tool also provides the legal basis for a data-driven algorithmic tool capable of being +automated at scale, merely because the latter serves the ‘equivalent’ purpose. This inappropriate invocation of +the argument from analogy overlooks the massively enhanced power of data-driven technologies, their opacity, +the susceptibility of humans to automation bias, and the instant reproducibility, transfer and storage of digital + +80 Karen Yeung, “Constitutional Principles in a Networked Digital Society” (2022), https://ssrn.com/abstract=4049141. + + + + + + +16 +data.81 For example, in a judicial review challenge to the South Wales Police use of live facial recognition +technology (FRT), the High Court reasoned that having one’s face subjected to analysis by live FRT was no +different from having one’s face photographed by an police officer in the course of his or her duty, which earlier +judicial decisions had confirmed may be lawfully undertaken by police on the basis of their common law powers +to prevent and detect crimes.82 Such reasoning reflects a failure to understand that the technological capacities +of contemporary data-driven, networked technologies are qualitatively different from an earlier generation of +pre-internet enabled tools, used for the same or similar purposes. The danger here is that the false assertion of +conformity may undermine the rule of law, while bypassing the need for public debate and deliberation about +whether the general public considers it appropriate for these technologies to be employed by law enforcement +authorities, and if so, on what terms and with what safeguards. Accordingly, we argue that the capabilities and +limitations of new technological tools in real-world contexts must be properly considered when scrutinising +claims that existing warrants of authority provide an adequate legal basis to authorise their use. If they expand +and extend dangers associated with their use, express statutory authorisation must first be provided, facilitating +open public debate and scrutiny rather than acquiesce in the expansion of the state’s coercive powers by +technological stealth. + +Secondly, the capacity of decision-support tools to operate automatically, at scale, yet in a highly opaque +manner, must be taken into consideration when evaluating the ‘proportionality’ of deploying any tool, even +assuming that it serves a legitimate and lawful purpose. We applaud the reasoning of the court in SyRI when +concluding that, although state authorities could legitimately employ data-driven tools to identify welfare fraud, +the gathering of administrative data from a wide range of unconnected sources to create highly detailed profiles +of individuals was disproportionate, violating the ECHR Article 8(1) right to privacy.83 In so doing, the Court +provided important judicial acknowledgement that analogue tools cannot be considered ‘equivalent’ to their +networked data-driven digital counterparts, even when employed to serve familiar and legitimate legal and +policy objectives, owing to qualitative differences arising from their capacity to scale, and the level of +intrusiveness that systematic data-collection across disparate contexts entails. + +Although algorithmic tools offer organisations the possibility of automating tasks at scale, this radically +magnifies their power and, in turn, the scope at which injustice and the abuse of power that they may generate. +Indeed, the recent UK Post Office Horizon scandal in which defective financial accounting software produced +by Fujitsu resulted in the wrongful conviction of 39 innocent individuals, considered the largest miscarriage of +justice in British history, powerfully illustrates how software can radically ‘scale injustice’ in a systemic yet +highly opaque manner which individuals may find practically impossible to contest.84 It highlights the urgent +need for systematic transparency, accountability and independent oversight to protect those subjected to +algorithmic evaluation within the criminal justice system, which our analysis reveals are sorely lacking. In +particular, institutional mechanisms of transparency and accountability are necessary in the decision to adopt +these tools in the first place, and the way in they are built, deployed, and evaluated. The UK currently has no +comprehensive, publicly available, inventory of decision-support tools used by criminal justice authorities that +provides even basic information about such tools, let alone supplying evidence of their beneficial and adverse +impacts. Indeed, there may be tools in use of which the public has no knowledge.85 Without meaningful +systematic oversight, algorithmic tools will continue being built and used in ways that automate and scale +injustice, enhancing the risks that governmental power will be abused. Systematic independent oversight is +urgently needed, including the creation and maintenance of a transparency register to help ensure that the +safeguards we have identified have been properly implemented.86 Such a register should, at minimum, include + +81 Skitka et al, “Automation bias and errors: Are crews better than individuals?” +82 R (Bridges) v South Wales Police and ors [2020] EWCA Civ 1058 at [54]-[85]. +83 SyRI Judgment at [6.78]-[6.85]. +84 Haroon Siddique, “Wrongly convicted Post Office workers to get up to £100,000 interim payouts” (The Guardian, 22 +July +2021), +https://www.theguardian.com/business/2021/jul/22/wrongly-convicted-post-office-workers-to-get-up-to- +100000-interim-payouts. +85 House of Lords Justice and Home Affairs Committee, “Technology rules?” 40: “On the most basic level, we cannot be +certain what technologies are being used for the application of the law in England and Wales.” +86 Also recommended by the House of Lords Justice and Home Affairs Committee, “Technology Rules?” at pp 45-46, +relating to “all advanced algorithms used in the application of the law that have direct or indirect implications for +individuals.” + + + + + + +17 +public information about the matters identified in section 4.2(d), as part and parcel of the ‘right to reasons’ +including: + +1. The legal authority upon which the tool may lawfully be developed and used, including the particular +decisions that the tool is intended to inform; +2. The configuration of error thresholds and other normative trade-offs made during tool-development; +3. Information concerning how those normative choices were made and by whom; +4. Input, test, and validation data used to produce the underlying mathematical model, and upon which the +tool relies to generate predictions; +5. Basic information about the outputs produced and what those outputs are taken to signify; +6. The type of machine learning used and why; +7. Measures concerning the tool-validation process employed, including relevant quality measures +(including accuracy, recall and sensitivity); +8. The extent to which the model’s predictions are supported by evidence demonstrating the causal +underpinnings of the claimed relationship between the features used and the phenomenon which the +model seeks to predict (or at least reasonable plausibility between those relations); +9. The intended ‘users’ of the tool, and the particular organisational and legal contexts in which they are +intended to be used; and +10. The range of specific procedural and other safeguards in place to guard against decisions taken on the +basis of error, illegality, irrationality or other forms of abuse of power, both individual and systematic. +5. +Conclusion + +This two-part paper has demonstrated how seemingly ‘technical’ choices made by developers when building +algorithmic tools for criminal justice authorities have serious constitutional implications which cannot be +reduced to issues of technical computational know-how. We have argued that, because technical developers +cannot reasonably be expected to have a proper understanding of public law principles, they must collaborate +closely with public law experts when deciding whether to employ decision-support tools for specific criminal +justice purposes, and if justified, to ensure they are configured in a manner that is demonstrably compliant with +public law principles and doctrine, including respect for human rights, throughout the tool-building process. It +is widely recognised that public law principles and safeguards apply to inform and constrain the use of weapons +and other enforcement tools and techniques employed by the police and other criminal justice authorities, such +as handcuffs, guns, tasers, teargas through to heightened surveillance, electronic tagging, and so forth. Yet there +has been a peculiar ongoing failure to recognise that these same principles and safeguards should be applied to +constrain and inform algorithmic decision-tools. Although public law scholars have highlighted how +algorithmic tools may be in tension with constitutional values, they have been surprisingly reluctant to argue, +and to demonstrate why constitutional principles operate as ‘red lines’ that mark out the boundaries of +acceptability. In contrast, by employing cross-disciplinary insights from public law and data science, we have +sought to demonstrate that unless and until these tools are designed and deployed in compliance with basic +constitutional principles and legal requirements, they should not be used.87 Systematic institutional oversight +mechanisms are urgently needed to ensure such compliance, otherwise algorithmic tools are likely to proliferate +in ways that violate individual rights, producing injustice and eroding public trust in the integrity of the criminal +justice system. + + +11.10.22 +12158 words + +87 Tobias D. Krafft, Katharina A. Zweig and Pascal D. König, “How to regulate algorithmic decision-making: A framework +of regulatory requirements for different applications” (2020) Regulation and Governance, 1-18. + diff --git a/P9E3T4oBgHgl3EQfyAsA/content/tmp_files/load_file.txt b/P9E3T4oBgHgl3EQfyAsA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6ee25401740757d8b0683a356343692eeddcd392 --- /dev/null +++ b/P9E3T4oBgHgl3EQfyAsA/content/tmp_files/load_file.txt @@ -0,0 +1,625 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf,len=624 +page_content='1 This manuscript has been accepted for publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' The official published version will appear in [2023] Public Law, forthcoming (published by Sweet & Maxwell) and made available in digital format on subscription from Westlaw UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' How do ‘technical’ design-choices made when building algorithmic decision- making tools for criminal justice authorities create constitutional dangers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' by Karen Yeunga and Adam Harkensb Part II 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Introduction The high-profile failure of automated digital decision tools by public authorities, particularly those that utilise some form of machine learning (ML), including the ‘robo-debt’ scandal in Australia1and the Dutch childcare benefits scandal,2 illustrate the scale and seriousness of the hardship and injustice that digital ‘solutions’ in government can produce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Reflecting on these systems, the UN’s previous Special Rapporteur on Extreme Poverty, Philip Alston has warned of the imminent dangers of ‘digital dystopia’3 highlighting the urgent need for safeguards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Although established public law principles could be mobilised to prevent mistakes and failures, they are yet to be effectively and systematically operationalised in the development, implementation, and oversight of public sector algorithmic tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' This two-part paper focuses on digital tools used by criminal justice authorities that purport assess the ‘risk’ posed by specific individuals to inform how they should be treated, although much of our analysis has wider applicability to the public sector more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' In Part I, we showed how ML-based automated digital decision-making and support tools (‘algorithmic tools’) are conventionally built and implemented without reference to their larger context of application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' We argued that, despite the ‘rights-critical’ nature of criminal justice decisions,4 algorithmic tools for public sector use are conventionally developed in ignorance of public law principles and the legal duties to which they give rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' For technical developers, these contextual considerations are conventionally regarded as irrelevant ‘noise’, informed by what we call a ‘contextual detachment mindset’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='5 As a result, vital institutional safeguards against the arbitrary or otherwise unjust exercise of power by public authorities are being circumvented, substantially enhancing the likelihood that public powers may be exercised unlawfully, creating injustice that is in practice difficult to detect and almost impossible to challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' In this Part II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' we demonstrate more precisely how choices made by technical developers during the design,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' construction and implementation of algorithmic tools implicate several legal duties that apply to the exercise of the decision-making authority that those tools inform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' including English administrative law doctrine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' human rights protections set out in the European Convention on Human Rights a Professorial Fellow in Law,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Ethics and Informatics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Birmingham Law School and School of Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' University of Birmingham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' We gratefully acknowledge funding support form VW Stiftung, Grant No: 19-0087 (2019- 2023) and for helpful feedback by Emma Ahmed-Rengers (particularly in comparing conventional statistics with ML approaches), Reuben Binns, Mireille Hildebrandt, Tobias Krafft, Winston Maxwell, Leandro Minku, Johannes Schmees, Georg Wenzelberger and Katharina Zweig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Karen Yeung drafted the text and devised the analytical framework, argument, paper structure and narrative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Adam Harkens undertook the in-depth case-studies and background research to the legal, scholarly and contextual detail supporting Yeung’s arguments and acted as a critical sounding board for her ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' An earlier version was presented by Karen Yeung to the Norwegian Association for Computers and the Law, The Knut Selmer Memorial Lecture, 23 November 2020 (Oslo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' b Post-doctoral Research Fellow, Birmingham Law School.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 1 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Henman, “Administrative justice in a digital world: Challenges and solutions” in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Tomlinson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Thomas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Hertogh and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Kirkham (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=') The Oxford Handbook of Administrative Justice (Oxford: OUP, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 2 Melissa Heikilla, “Dutch scandal serves as a warning for Europe over risks of using algorithms” (29 March 2022, Politico EU), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='politico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='eu/article/dutch-scandal-serves-as-a-warning-for-europe-over-risks-of-using-algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 3 Philip Alston, ‘Report of the Special Rapporteur on Extreme Poverty and Human Rights’ A/74/48037 (New York: United Nations, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 4 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', such decisions may interfere with the legal and fundamental rights of affected individuals: see Part I section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 5 See K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='Yeung and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Harkens, “How do ‘technical’ design-choices made when building algorithmic decision-making tools for criminal justice authorities create constitutional dangers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Part I” [2022] Public Law, forthcoming at section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 2 (ECHR) and incorporated by the Human Rights Act (HRA), data protection laws arising under that Data Protection Act 2018, and the so-called Public Sector Equality Duty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='5 In so doing, we seek to move beyond existing academic inquiry that has tended to be rather general and abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' This high-level of generality is problematic because,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' as legal scholars commissioned by the Administrative Conference of the United States (ACUS) to review federal agency use of algorithmic tools have observed: “…much,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' if not most,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' of the hard work regulating algorithmic governance tools will come not in the constitutional clouds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' but rather in the streets of administrative law.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='6 We begin with a brief account of three algorithmic tools that purport to assess the ‘risk’ posed by individuals (‘i-RATS’) currently (or until recently) in use by criminal justice authorities: two tools used in England - the London Gangs Matrix and the Durham Constabulary’s HART tool,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' and the SyRI tool formerly used in the Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' We then examine the intersection between public law and data science perspectives by adopting the lens of ‘algorithmic regulation’, drawing selectively from these three i-RATs, to demonstrate how particular abstraction decisions involved in algorithmic model-building directly implicate constitutional principles at each stage of the development process yet are conventionally and routinely ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' We argue that algorithmic tool- developers, and the authorities who commission and implement them, have failed to recognise, or understand, the constitutional and legal implications of these technical choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Hence, algorithmic tools are being employed by criminal justice authorities in ways that unjustifiably violate constitutional principles and the specific legal duties to which they give rise, significantly enhancing the risk and magnitude of injustice and abuses of power that can arise from their use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' The third and fourth sections consider the implications of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' We suggest that constitutional principles should be mandatory requirements forming an essential part of the design-brief which those who build algorithmic tools to inform criminal justice decision-making must adhere to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' because technical developers cannot be expected to understand nor properly apply public law principles and duties,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' we argue that they must collaborate closely with legal experts when deciding whether to deploy these tools for specific criminal justice purposes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' and if justified,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' to ensure that they are configured in a manner that is demonstrably compliant with public law principles (including respect for human rights) and all applicable legal duties throughout the tool-building process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' If such compliance cannot be demonstrated, they should not be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Given that lawyers unfamiliar with algorithmic model-building may struggle to understand how such principles and duties are implicated in technical design-choices, we also offer several practical recommendations, highlighting several ‘detachment practices’ (that is, practices in which legally and constitutionally relevant matters are conventionally ignored by technical developers) that must be avoided when algorithmic tools are developed for use by criminal justice decision-makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Finally, we outline a series of urgently-needed systematic legal reforms to help establish and maintain public trust in criminal justice decision-making in an increasingly algorithmic age, followed by a brief conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Three algorithmic risk assessment tools used to categorise and ‘flag’ individuals We begin with a brief account of three i-RATS used in criminal justice settings based on knowledge gleaned largely from publicly-available documents: The London Gangs Matrix, the Durham ‘HART’ tool, and the Dutch SyRI tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Although their technical dimensions and intended purposes vary significantly (to reduce gang violence, to enhance the effectiveness of offender ‘rehabilitation’, and more efficient identification of social welfare fraudsters respectively), they all produce an algorithmically generated assessment of an individual’s ‘risk’ for use by front-line decision-makers when deciding what action to take against those individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' It is important to emphasise, however, that their effectiveness in achieving their claimed criminal justice purposes has not been systematically evaluated, let alone proven, and this, as we shall see, is a matter of considerable constitutional importance given that they entail prima face interferences with human rights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='8 5 Equality Act 2010, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 6 David Freeman Engstrom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', “Government by algorithm: Artificial intelligence in federal administrative agencies” (Report submitted to the Administrative Conference of the United States, 2020), 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', consider the ‘Waterproof’ project referred to in the SyRI case, discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='1 of Part I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='1 The London Gangs Matrix The Gangs Matrix was created by the London Metropolitan Police Service (MPS) following the London riots in August 2011 to help reduce ‘street-focused’ gang violence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' It provides police officers, via their internet- enabled smart devices, with a dynamic digital dashboard and database that lists individuals identified as potentially ‘at risk’ of involvement in gang violence (called ‘Gang Nominals’) together with a ‘harm score’ generated via an algorithmic assessment tool believed to be created using ML techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='9 The MPS claims that the Gangs Matrix (a) helps police officers identify and assess in real-time the ‘risk of violent re-offending’ of a Gang Nominal by classifying them as ‘high’ (red), ‘medium’ (amber) or ‘low’ (green);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' and (b) helps allocate MPS enforcement resources, prioritising those deemed most dangerous while others are diverted and offered alternative support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Individuals can be included on the Matrix if identified as a potential gang member by a police officer or partner agency representative (including local councils,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' schools,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' and health services) and this has been corroborated by ‘reliable intelligence from more than one source.’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Intelligence deemed ‘reliable’ could include the fact that the individual has been observed associating with,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' being related to,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' or subject to a stop and search report together with a Gang Nominal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' or even sharing social media content that refers to a specific gang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Neither details of the model used to calculate the ‘harm score’ nor the data used to create it have been published, although the MPS states that factors contributing to the score include previous history of violence (weighted by seriousness of offence), violence or weapons intelligence in the previous six months, judgments of a local gang unit intelligence manager, and partner organisations’ assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' How these factors are weighted or utilised to calculate an individual’s harm score is not publicly known: the MPS merely state that it is based on a ‘complex scoring system’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='7 Individuals listed on the Matrix may be subjected to heightened police surveillance, often leading to pre-emptive stop and search, arrest for minor offences, anti-social behaviour injunctions and/or criminal behaviour orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Although not originally intended, the Matrix is also allegedly used in evidence to support the prosecution of gang-related offences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Individuals listed on the Matrix have been subjected to police stop and search more frequently than the general population, with subsequently reduced levels of police attention for those whose details are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Individuals identified as potential ‘victims’ of gang crime and other vulnerable persons may also be included in the Matrix, albeit accompanied by a ‘zero harm’ score, as are individuals designated as Gang Nominals with no previous criminal convictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='2 The Harm Assessment Risk Tool (HART) HART is an algorithmic tool that, until recently,8 was used by Durham Constabulary to help custody officers decide whether arrested individuals should be offered an opportunity to participate in its ‘Checkpoint’ rehabilitation program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' HART was developed collaboratively by Durham Constabulary and Cambridge University researchers using ‘custody event data’9 drawn from the Constabulary’s custody management IT systems for the five years to 31 December 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='10 Considerably more information about the HART tool is available than is typically the case for public sector decision-support tools, thanks to Sheena Urwin (then Head of Criminal Justice at Durham Constabulary), who wrote her Master’s thesis under the supervision of Dr 9 Amnesty International United Kingdom Section (“Amnesty International”), “Trapped in the Matrix: Secrecy, stigma, and bias in the Met’s Gangs Database” (May 2018) 20 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='amnesty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='uk/files/reports/Trapped%20in%20the%20Matrix%20Amnesty%20report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='pdf, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 7 MOPAC, “Review of the MPS Gangs Matrix” at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='london.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='uk/sites/default/files/gangs_matrix_review_- _final.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='pdf, 20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Amnesty International, “Trapped in the Matrix,” 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 8 House of Lords Justice and Home Affairs Committee, “Technology rules?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' The advent of new technologies in the justice system” (HL 2021-2022, 180-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 9 A ‘custody event’ refers to the disposal decision taken by the custody officer following arrest at the end of the first custody period,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' either to grant bail,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' remand D in custody,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' taken no further action,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' administer an out of court disposal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' or prosecute the subject (with a decision to bail): Sheena Urwin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' “Algorithmic forecasting of offender dangerousness for police custody officers: An assessment of accuracy for the Durham Constabulary model”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Research Presented as for the purposes of gaining a Master’s Degree in Applied Criminology and Police Management at Cambridge University (2016) at http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='crim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='uk/alumni/theses/Sheena%20Urwin%20Thesis%2012-12-2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='pdf, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 10 Urwin, “Algorithmic forecasting of offender dangerousness,” 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' See Teresa Scantamburlo, Andrew Charlesworth and Nello Cristianini, “Machine Decisions and Human Consequences,” in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Yeung and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Lodge (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' ), Algorithmic Regulation (Oxford: Oxford University Press, 2019) which explains the technical development of the HART tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 4 Geoffrey Barnes - one of the academics credited with designing HART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='11 Urwin’s thesis,12 which she published online, explains that HART used thirty-four risk predictors based on data about the arrested person at the time of arrest, combined with data from Durham Constabulary’s pre-existing records, created using a random forest ML modelling technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' HART purported to calculate the arrestee’s risk of committing an offence in the subsequent two years, described as a prediction of ‘offender dangerousness’ (a misnomer: those arrested have not been convicted and were therefore wrongly described as ‘offenders’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Individuals predicted as likely to commit a ‘serious’ offence (defined as an offence involving violence), a non-serious offence, or no offence, were classified as ‘high’, ‘moderate’ or ‘low’ risk respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='13 Only those receiving a ‘moderate’ prediction were eligible for Checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Checkpoint participants are provided with an ‘out-of-court disposal’ which involves signing a 4-month deferred prosecution contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Provided they meet the contract conditions throughout (including refraining from offending)14 participants avoid formal prosecution and a criminal record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Individuals could not be diverted into Checkpoint unless the custody officer was satisfied that there was sufficient evidence to charge, in line with Crown Prosecution Service requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='3 SyRI SyRI was an ML-based decision-support tool used by several Dutch public authorities to help detect tax and social security fraud until it was declared unlawful following a successful judicial review challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='15 SyRI was built from data extracted from individual’s administrative records held by government agencies to generate individual risk profiles to identify suspected welfare fraud (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' those with “an increased risk of irregularities” when compared to the target population).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='16 Potential risk indicators were based on demographic data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', name, address, and associated administrative data), administrative fines and sanctions, debt burdens, social benefit, and tax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' If algorithmically flagged, the individual’s profile was then investigated by the Ministry of Social Affairs and Employment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Little is publicly known about the substantive interventions that followed once individuals had been notified that they had been flagged for investigation, but reportedly include administrative sanctions/fines, mandatory participation in civic education programmes, or in sufficiently serious cases, referral to the Public Prosecution Service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='17 In short, SyRI offered Dutch authorities a powerful technological means for monitoring individuals in real-time, linking vast swathes of personal, economic and administrative data collected from records held by multiple governmental organisations to inform how public authorities exercise their investigative and enforcement discretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Understanding tool-building through the lens of ‘algorithmic regulation’ In Part I, we explained how statistical prediction models built using ML techniques conventionally entails making abstraction decisions that intentionally ignore context-specific features of the real-world domain in 11 Geoffrey Barnes was then an Affiliated Lecturer at Cambridge University’s Institute of Criminology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' see Josh Jacobs ‘The radical idea to reduce crime by policing less, not more’ (Wired, 10 March 2021) https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='wired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='uk/article/evidence-based-policing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Information about HART is available from an academic journal article jointly written by Barnes, who was responsible for its technical design, and Urwin, who occupied a senior role in the Durham Constabulary at the time of its use - along with legal scholars Oswald and Grace: Marion Oswald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', “Algorithmic risk assessment policing models: lessons from the Durham HART model and ‘Experimental’ proportionality” (2018) 27(2) Information & Communications Technology Law 223-250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Although the paper claims to offer a ‘critique’ of the HART tool, we do not understand why the authors declared that there was “no potential conflict of interest” at 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 12 Urwin’s thesis sought to assess the validated accuracy of the HART model: “Algorithmic forecasting of offender dangerousness.” 37;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Oswald et al, “Algorithmic risk assessment policing models,” 229-230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 13 Urwin, “Algorithmic forecasting of offender dangerousness,” 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 14 Contracts can include up to five conditions: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', refraining from re-offending, participating in restorative processes with victims, attending mental health or substance abuse therapy sessions, or wearing a GPS tag: Kevin Weir, Gillian Routledge & Stephanie Kilili, “Checkpoint: An Innovative Programme to Navigate People Away from the Cycle of Reoffending: Implementation Phase Evaluation” (2019) 15 Policing: A Journal of Policy and Practice 1-19, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 15 NJCM et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' and FNV, Rechtbank den Haag, ECLI: NL: RBDHA: 2020:1878 (“SyRI Judgment”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 16 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', the Tax and Customs Administration, municipal authorities, the Public Prosecution Service, the police, and the immigration service: SyRI judgment, [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 17 SyRI Judgment at [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' [6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 5 which the tool will operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' For algorithmic tools intended to inform criminal justice decision-making, ignoring these context-specific features includes ignoring the larger constitutional backdrop against which these decisions are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' We now demonstrate more precisely how particular design-choices in algorithmic tool- building implicate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' and must be constrained by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' public law principles and legal duties by adopting the lens of ‘algorithmic regulation.’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='18 This analytical approach begins from the perspective of the data scientist commissioned to build an algorithmic tool for a specific organisational purpose,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' tracing the steps and design choices involved and then proceeds to critically investigate the points of contact between the algorithmic tool and the social world of its deployment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' particularly their impacts upon those subjected to algorithmic evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Although this second stage allows many different analytical perspectives,19 we focus on the constitutional principles that a commitment to constitutionalism would impose as non-negotiable requirements – or ‘parameters’ to use more familiar computer science terminology – that should condition those choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' By decomposing ML tool-building into four-steps, we demonstrate in concrete terms how design choices implicate constitutional principles and the specific legal duties that operate to inform and constrain those choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' For simplicity, we consider a specific set of decisions made under conditions of uncertainty involving some kind of classification task: such as distinguishing ‘spam’ emails from legitimate (‘ham’) emails, to identify whether a patient is a diabetic or not, or to evaluate whether a person currently serving a custodial sentence for a criminal offence will re-offend on release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Imagine an organisation whose front-line decision-makers are routinely required to make decisions of this kind, and commissions the building of an algorithmic tool to assist them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' To create such a tool, the algorithmic model-building process proceeds in four steps20: Step 1: Identify suitable input data to serve as ‘ground truth’ for model-building;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Step 2: Create an algorithmic model by applying ML techniques to generate predictions concerning future outcomes on unseen data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Step 3: Test and validate the model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' and Step 4: Encode the model into a software program capable of communicating its outputs via a digital interface to create a convenient digital tool to assist front-line workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='1 Step 1: Obtain a suitable dataset to serve as ground-truth to train a prediction model First, the developer must identify a dataset to serve as ‘ground truth’ in which each data item possesses the relevant feature of interest upon which to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='21 Properties used to describe the item to be classified are often called ‘features’ while the classes assigned to each item are called ‘labels’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='22 The relevant features of any given item depend upon the phenomenon which the developer chooses to rely on as the basis for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' For our three examples, the features might refer respectively to: words used in the email, a set of clinical indicators, or a person’s criminal history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' The labels applied to the training data comprised of these items might be respectively labelled ‘spam’ or ‘ham’, ‘diabetic’ or ‘healthy’, or ‘safe’ or ‘dangerous.’ A machine learning algorithm may then be applied to this labelled training data to build a mathematical function (a ‘classifier’) that assigns a class label (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', ‘spam’ or ‘ham’) to any object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', emails, patients, prisoners) that has not yet been labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='23 The resulting classifier can be used to make predictions on unseen data, helping to inform (or even to automate) decisions for a specific organisational purpose, such as the creation of an 18 Karen Yeung, “Algorithmic regulation: A critical interrogation” (2018) 12 (4) Regulation and Governance 505-523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 19 See Lena Ulbricht and Karen Yeung “Algorithmic regulation: A maturing concept for investigating regulation of and through algorithms” (2021) 16(1) Regulation and Governance 3-22 on the ‘thin’ nature of algorithmic regulation as an analytical frame and the ‘thicker’ variety of conceptual lenses it allows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 20 For a basic explanation of how ML-based models utilising supervised learning are created, see Part I section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 21 David Lehr and Paul Ohm, “Playing with the Data: What Legal Scholars Should Learn About Machine Learning” [2017] 51 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Davis L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 654 – 717, 676;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Hildebrandt, “Algorithmic regulation and the rule of law” (2018) 376 Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' A 1-11, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 22 Lehr and Ohm, “Playing with the Data,” 665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 23 Scantamburlo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', “Machine Decisions and Human Consequences,” 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 6 automated email-filtering tool, a screening tool to help identify patients likely to be (or become) diabetic, or to assist a parole board deciding whether to grant a prisoner early-release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' This model-building process relies on several critical assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' First, that the training data serves as appropriate ‘ground truth’, a matter to which we will return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Secondly, that historic data provides a reliable guide to future instances of the phenomenon it is taken to represent: this assumes that the relevant practices and their surrounding context remain stable and unchanged, and that the training data was collected in a context relevantly similar to the proposed context of deployment: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', new forms of spam might not be accurately identified by a spam-predictor that has not included training data with that kind of spam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' However, the conventional training of computer scientists might not extend to matters concerning the collection of training data and/or its relationship to real-world phenomenon which their models seek to predict beyond a narrow, relatively limited set of technical requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' That said, the work of data scientists involves attending to the ‘quality’ of datasets to render them useful for algorithmic modelling, although data ‘quality’ is context dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Poor quality data includes ‘dirty’ data, suffering from problems that later hinder the quality of a given algorithmic model thus requiring human ‘cleaning’4 to render the data suitable for model-building,24 or the data may be incomplete, noisy, contain significant outliers, infused with errors (caused by a variety of human and computer fallibilities, such as a failure in code, or a failure in human data entry) or formatted in a manner that is not processable by modelling software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' As the European Fundamental Rights Agency (FRA) has observed, the use of ‘low quality’ data to produce prediction models that produce ‘low quality’ outcomes to inform real-world decision-making can lead to violations of fundamental rights, particularly rights to privacy and data protection25 while negatively affecting the right to an effective remedy, lamenting the failure of many textbooks and articles dealing with data science to overlook these crucial aspects of data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='26 Identifying and selecting the appropriate input and training data requires active involvement and normative judgement by the data scientist to ‘wrangle’ a dataset into a useable format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' However, even if technical developers regard themselves as professionally responsible for some matters of data quality, this might not extend to the legality of data collection and processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='27 For algorithmic tools intended to assist criminal justice authorities, routinely collected administrative data is often used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Yet even assuming that such data can be lawfully collected, it does not necessarily follow that it can be lawfully used to build algorithmic tools28 for at least two reasons: (a) Unlawful data collection and processing Firstly, privacy laws and contemporary data protection laws restrict the collection and processing of ‘personal data’ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', data pertaining to an identified or identifiable individual) in significant ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='29 Algorithmic tools used for criminal justice purposes have been found to violate these laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='30 For example, the Hague District Court ruled that the state could lawfully create and employ data-driven technologies to identify fraudulent 24 Tarleton Gillespie, ‘The Relevance of Algorithms’ in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Gillespie, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Boczkowski and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Foot (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=') Media technologies: Essays on Communication, Materiality, and Society (MIT Press, 2014), 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 25 European Union Agency for Fundamental Rights, “Data quality and artificial intelligence: mitigating bias and error to protect fundamental rights” (11 June 2019) https://fra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='eu/sites/default/files/fra_uploads/fra-2019-data-quality- and-ai_en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 26 European Union Agency for Fundamental Rights, “Data quality and artificial intelligence,” 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 27 Andrew Selbst, Suresh Venkatasubramanian and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Elizabeth Jumar, “The legal construction of black boxes: How machine learning practice informs foreseeability” (2021) Paper presented at the We Robot 2021 conference, https://werobot2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='com/wp-content/uploads/2021/08/Kumar_et_al_Legal-Construction-of-Black-Boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' (cited with the kind permission of the authors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 28 It might be lawful to use some data train a model, but not to use the same set of features about a person to make/inform a decision about them using the model in the real world, or vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' See ICO guidance on why controllers need distinct lawful bases for development and deployment: Information Commissioner’s Office, “What do we need to do to ensure lawfulness, fairness, and transparency in AI systems?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' at https://ico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='uk/for-organisations/guide-to-data- protection/key-dp-themes/guidance-on-ai-and-data-protection/what-do-we-need-to-do-to-ensure-lawfulness-fairness- and-transparency-in-ai-systems/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='q=profiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' We are grateful to Reuben Binns for drawing this to our attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 29 Article 4(1) of the EU’s General Data Protection Regulation 2018 provides that ‘personal data’ applies only to information ‘relating to an identified or identifiable natural person (‘data subject’).’ See Case C-582/14 Breyer v Germany [2014] ECLI:EU:C:2016:779).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 30 Although there is room for debate about whether the resulting model could be considered unlawful per se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 7 activity, SyRI was disproportionately intrusive and thus not legally justified as ‘necessary in a democratic society’, violating the Art 8(1) right to private life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='31 In a related vein, the UK Information Commissioner’s Office (ICO) ruled that by naming identifiable individuals and labelling them ‘Gang Nominals’, the Gangs Matrix violated data protection laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Although the data parsed by the Matrix to generate individual harm scores is drawn primarily from administrative data lawfully collected by the London MPS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' it relied upon the excessive collection of personal data including ‘gang association data’ on informal unregulated lists,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' entailed unregulated data sharing between partner agencies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' and included victim data in the Matrix without distinguishing them from suspected ‘Gang Nominals.’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='32 (b) Increasing the likelihood of the unlawful exercise of public decision-making authority Secondly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' developers may regard the legal duties applicable to public authority decision-making as matters as outside their problem-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' But without a proper understanding of these duties, the tools they build may facilitate unlawful decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Moreover, decisions attached to a particular office (including those taken by criminal justice officials, such as the police) are typically accompanied by specific legal duties, including administrative law duties that restrict the lawful exercise of discretionary power in the hands of public officials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' For example, British administrative law requires that public power be exercised in accordance with the purpose for which that power was conferred, and only on the basis of ‘relevant considerations,’ while ‘irrelevant’ considerations must be excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='33 Accordingly, data used to train the algorithmic model must generate outputs that are legally ‘relevant’ to the decisions those outputs inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' But if the phenomena that the training data is taken to represent does not conform with legal requirements, then the resulting model may generate predictions that are legally ‘irrelevant’ to the decision at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='34 This might occur, for example, if the training data used is not a valid or lawful indicator of the predicted phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' So, if legislation requires consider of variable A only, it is not legally permissible for the decision-maker to consider variable B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='35 Yet if no ‘ground truth’ data concerning the matter which a public authority is legally required to consider when making particular decisions is available, data scientists might utilise data-sets they regard as a ‘proxy’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='36 This creates serious risks that the model’s outputs will be legally irrelevant to the decision at hand resulting in decision-making that is ultra vires and therefore unlawful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' For example, consider a police officer in England or Wales who is legally required to decide whether an arrested person in custody (D) should be released without charge (or under continuing investigation), charged and publicly released on bail, or charged and detained in custody on remand prior to trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='37 Continued detention on remand pending trial is only lawful if the conditions of s 3(6) of the Bail Act have been met and authorised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' This requires establishing (among other things) that detention is deemed ‘necessary’ to secure D from absconding, to prevent D committing an offence while on bail, or to protect D or another person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='38 It is in making 31 ECHR Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 32 Information Commissioner’s Office, “Enforcement Notice to the Commissioner of Police of the Metropolis” (13 November 2018), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='police.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='uk/SysSiteAssets/media/downloads/force-content/met/about-us/gangs-violence- matrix/ico-enforcement-notice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 33 R v Secretary of State for the Home Department, ex parte Venables and Thompson [1998] AC 407 34 Marion Oswald, ‘Algorithm-assisted decision-making in the public sector: Framing the issues using administrative law rules governing discretionary power’ [2018] 376 Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' A 1-20, 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 35 The matter may be arguable if variable A can be consistently and reliably inferred from variable B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Difficulties can arise in evaluating the lawfulness of algorithmic models that have been generated through the use of datasets from which sensitive attributes (for example, such as gender or race) have been removed, because such sensitive attributes may well be readily inferred from other features in the data: see A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Roth and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Kearns, The Ethical Algorithm: The Science of Socially Aware Algorithm Design (Oxford: OUP, 2019), 74-84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 36 See discussion of the limitations of proxy data within the contexts of education, social care and criminal justice in Abigail Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Jacobs and Hanna Wallach, “Measurement and Fairness” (2021) FAccT’ 21 Proceedings of the 2021 ACM Conference on Fairness, Accountability and Transparency, 375-385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 37 For a more expansive illustration of this decision process in the context of the use of algorithmic tools, along with applicable laws, see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' of Part I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 38 Bail Act 1976, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='3(6): the decision whether to refuse bail and to remand D pending trial is made by a magistrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 8 these kinds of determinations that US criminal justice authorities have enthusiastically embraced algorithmic tools that purport to assess the risk that an individual will abscond (‘flight risk’) or commit a criminal offence if released (so-called ‘recidivism risk’),39 although we are not aware of their use by British courts for these purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' In assessing the necessity of retaining D in custody for the purposes permitted by the Act, some kind of assessment of the likelihood that the individual will commit a criminal offence, engage in self-harm, or harm another person during the period prior to trial is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' To create an algorithmic model capable of providing such predictions, ‘ground truth’ data would consist of a comprehensive historic data set identifying the entire universe of crimes, and of individuals who have engaged in criminal conduct or otherwise harmful conduct directed at others or themselves for the geographic region in question over a substantial yet recent time-period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Yet because not all crimes or harmful actions are identified, reported, and recorded, and many of those who commit crimes and/or harms evade apprehension and conviction, in addition to a lack of systematic data concerning self-harming activities falling short of suicide, ground truth data about the commission of crime and harm across a population, even within a more narrowly defined geographic area, is not available and, indeed, is likely impossible to collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' This has not, however, prevented developers from building so-called ‘recidivism risk’ predictors using arrest data, treating arrest as a ‘proxy’ for crime committed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Arrest data is, however, an inadequate, highly misleading indicator of crime because not all arrested persons are charged, those who are charged may not be convicted, and many crimes are committed for which no arrests are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Accordingly, the mere fact of arrest is not an acceptable proxy for the commission of criminal offence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Thus, by failing to consider what the training data actually signifies, the tool’s outputs may not offer meaningful indications of the phenomenon it purportedly predicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Accordingly, the tool’s outputs may be ‘legally irrelevant’ to the matter the public official must decide and therefore cannot lawfully be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' In the above example, it would not be legally permissible to remand in custody an arrestee charged with a criminal offence solely on the basis of an algorithmic tool that predicts the likelihood that D will be re-arrested because the legal significance of a lawful arrest differs very significantly from that of a criminal conviction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='40 But it is on the basis of arrest data, for example, that the HART tool was developed, which is described as generating predictions about whether an individual is likely to ‘commit a criminal offence’ within a two year period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' At the very least, these tools should be properly described as ‘arrest predictors’ rather than mislabelling and misrepresenting them as predictions of future criminal offending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='41 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='2 Step 2: Apply a machine learning algorithm to create a prediction model Armed with labelled ‘ground truth’ data, ML software is then applied to train and build an algorithmic model (a ‘classifier’) that assigns a class label (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', ‘spam or ham’, ‘diabetic or healthy’, ‘safe or dangerous’) to unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Advanced ML techniques enable the generation of more powerful and sophisticated mathematical models by analysing larger, more complex, highly dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Because many ML methods are available, technical developers must choose between them (and may experiment with several alternatives) including regression models that generate a score for each candidate or classification models that allocate candidates into classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' One parameter of great importance, particularly for algorithmic tools used in rights-critical contexts, concerns the configuration of the model’s ‘error thresholds.’42 Predictions generated using ML models are inherently probabilistic: prediction errors are therefore unavoidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' In this context, errors are defined as the failure of the model to generate an accurate prediction on unseen data, either in the form of a Type I error (false positive) or Type II (false negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='43 Good data scientists also recognise the vital importance of attending to the distribution of Type I and Type II errors in building algorithmic models, requiring careful consideration of 39 Kathrin Hartmann & Georg Wenzelburger, “Uncertainty, risk and the use of algorithms in policy decisions: a case study on criminal justice in the USA” (2021) 54 Policy Sciences 269-287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 40 Virginia Eubanks, Automating Inequality: How High-Tech Tools Profile, Police and Punish the Poor (New York: St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Martin’s Press, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' David Robinson “The Challenges of Prediction: Lessons from Criminal Justice.” (2017) 14 Journal of Law & Policy for the Information Society Robinson, 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 41 Robinson, “The Challenges of Prediction,” 160-161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 42 Steven W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Knox, Machine Learning: A concise introduction (Chichester: Wiley, 2018), 15-32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Spiegelhalter, The Art of Statistics: Learning from Data (Pelican, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 43 Spiegelhalter, The Art of Statistics: Learning from Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 9 their real-world consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' For example, consider the consequences of errors produced by an algorithmic tool that evaluates skin melanoma images to predict whether a melanoma is cancerous or benign, intended for at-home diagnosis to help individuals decide whether to seek medical advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' A Type II error (false negative) is considerably more serious: a person who mistakenly believes that the melanoma is benign may not seek further medical treatment, allowing the cancer to develop untreated, with potentially fatal consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' In contrast, a Type I error (false positive) is likely to prompt that person to seek medical assistance unnecessarily, assuming that the melanoma is subsequently diagnosed correctly by the clinician as benign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='44 Careful consideration of the real-world consequences of different error-types is particularly important when determining the cut-off point at which point the model assigns a classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' As Scantamburlo et al explain, when ML techniques are used to build binary classifiers that can assign an item to one of many possible categories (such as ‘spam’ or ‘ham), many (although not all) involve in a two-stage process: first, a real valued score is computed for the item to be classified, and secondly, that score is compared with a cut-off point whereupon the item is assigned to a class depending on whether it exceeds the cut-off point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='45 The real valued score could informally be thought of as a probability, though it is not necessarily a formal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' In some (but not all) cases, changing the threshold results in a trade-off between false positives (Type I error) and false negatives (Type II error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' In the example of skin-cancer predictor, this can be captured by notions of ‘sensitivity or recall’ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', the true positive rate is the proportion of true positives in relation to the aggregate of true positives + false negatives) and ‘specificity’ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', the true negative rate is the proportion of true negatives in relation to the aggregate of true negatives + false positives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' For example, for our melanoma classifier, because the consequences of mistakenly classifying a melanoma as benign when it is in fact cancerous (Type II error) is far more serious than mistakenly classifying a benign melanoma as cancerous (Type I error), the cut-off point should be configured to minimise Type II errors, even at the cost of increasing Type I errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' These examples also demonstrate that the real-world consequences of error are context-dependant: a Type I error in ‘recidivism risk’ prediction is far more serious for an affected individual than a Type I error by an automated spam-filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' As a matter of constitutional principle, algorithmic tools that inform how an individual will be treated by criminal justice authorities must be configured to distribute the risk of error in a manner that respects due process rights, including the presumption of innocence protected under ECHR Art 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' These procedural rights are rooted in the state’s duty to demonstrate respect for persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' It demands that those adversely affected by the decision of a public authority, particularly criminal justice authorities wielding the coercive power of the state, are entitled to be informed of, and to challenge those decisions, particularly given the ever-present danger of mistakes in decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' As a matter of administrative law, the specific procedural requirements (and concomitant duties) that the right to due process demands is highly context-dependent, in which the decision’s impact upon the affected individual’s rights, interests and legitimate expectations has great importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='46 Hence a person whose driver’s licence application is refused enjoys a fairly limited set of procedural rights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='47 while a person charged with committing a very serious criminal offence is entitled to an extensive suite of procedural rights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' typically including rights to free legal representation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' to the application of strict rules of evidence and procedure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' and to the presumption of innocence in which the burden of providing the accused’s guilt ‘beyond reasonable doubt’ is placed firmly on the prosecution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='48 This high standard of proof, which is far more demanding than the ‘balance of probabilities’ standard applicable to civil cases, produces a legal system systematically designed to minimise Type I errors (false positives) while producing more Type II errors (false negatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' This means guilty defendants may avoid conviction because the prosecution has failed to prove guilt ‘beyond reasonable doubt.’ Although such outcomes are deeply regrettable, these errors are nevertheless widely accepted in modern western European legal systems as worth incurring to avoid the far more egregious injustice associated with Type I errors—that is, wrongfully convicting an innocent person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Particular care is needed for 44 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', the patient will not then be subjected as very invasive, unnecessary interventions due to an incorrect diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' See for example Karsten Juhl Jørgensen, Peter C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Gøtzsche, Mette Kalager, and Per-Henrik Zahl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' “Breast Cancer Screening in Denmark: A Cohort Study of Tumor Size and Overdiagnosis” [2017] 166(5) Ann Intern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 313-323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 45 Scantamburlo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', “Machine Decisions and Human Consequences,” 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 46 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Galligan, Due Process and Fair Procedures: A Study of Administrative Procedures (Oxford: OUP, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 47 Including rights to be tested by an unbiased examiner, to be given a reasonable opportunity to demonstrate that their driving skills meet the requisite legal standard, and to be provided with reasons for any refusal of a license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 48 ECHR, Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 10 prediction models created by ML techniques which exacerbate the dangers associated with mistakes because they are produced on the basis of correlations in the underlying training data rather than on scientifically established causal relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='49 To demonstrate how human rights should condition algorithmic tool-design, consider again the decision to detain or release a person charged with a criminal offence (D) pending trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' To detain a person against her will for extended periods entails a grave and serious denial of liberty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' If these decisions are to be informed by predictions that purport to classify an individual as ‘dangerous’ or ‘harmless’ such that the former is held in remand while the latter released, then falsely classifying a harmless person as ‘dangerous’ (Type I error) will entail a serious interference with that person’s right to liberty under Article 5 ECHR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='50 However, falsely classifying as ‘harmless’ a person (‘D’), who is in fact dangerous, will result in release, placing the public at risk that D will commit a serious crime during the pre-trial period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Nevertheless, if we are to take the presumption of innocence and the individual’s right to liberty seriously, this necessitates treating false positives (convicting the innocent) as far more serious than false negatives (failing to convict the guilty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' As a matter of constitutional principle, this relative weighting must be reflected in the construction of the underlying mathematical model, although the human decision-maker might justifiably override the algorithmic recommendation if release may expose specific, vulnerable individuals to a substantial risk of domestic violence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Although this means the public systematically bears a proportionately greater risk that a dangerous person will be released pending trial, constitutional democratic societies accept that this is the inescapable cost of upholding individual rights and freedoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' If the model were to be configured otherwise, this may unlawfully violate the arrestee’s Article 5 ECHR right to liberty and security, including freedom from arbitrary detention, and will also increase the likelihood of further violation of her Article 6 ECHR rights, including the right to be presumed innocent and the right to a fair trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Despite the importance of these technical choices, there is little public information concerning how the risks of Type I and II error are configured into algorithmic tools used by criminal justice authorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Of our three case studies, public information was only available in relation to the configuration of error thresholds for the HART tool, thanks to Urwin’s published Master’s thesis and the account provided in a jointly authored paper with Barnes, Oswald and Grace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='51 Yet those accounts reveal a disturbing failure to demonstrate any recognition that an arrested person is entitled to be presumed innocent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' stating that error thresholds were configured on the basis that “it is worse to misclassify a dangerous offender as harmless than to erroneously classify a harmless individual as dangerous.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='52 Although this may reflect the perspective favoured by the general public,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' it fails to recognise that such a choice entails an unjustified violation of the affected individual’s right to be presumed innocent (and hence harmless).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='53 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='3 Step 3: Model testing and validation Having generated a mathematical model, computer scientists then conventionally assess its ‘accuracy,’ understood as how well it correctly predicts outcomes from a set of unseen historic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Binary classification 49 See section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='1 of Part I for further discussion of the constitutional acceptability of using ML models to make predictions about individuals in rights-critical contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 50 ECHR, Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 51 Oswald et al, in “Algorithmic risk assessment policing models,” state at 236: ‘The HART model represents a real example of a value-judgement built into an algorithm, so requiring a ‘trade-off’ to be made between false positives and false negatives in order to avoid errors that are thought to be the most dangerous: in this context, offenders who are predicted to be relatively safe, but then go on to commit a serious violent offence (high risk false negatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' As a consequence, high risk false positives have been deliberately made more likely to result.’ 52 Urwin, “Algorithmic forecasting of offender dangerousness.” 53 Further, HART’s error thresholds were unjustified because they were insufficiently tailored to the policy purpose for which it was introduced to serve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Neither high risk false positives, where the individual was actually low risk, nor high false negatives where the individual was actually high risk, result in any change in the substantive recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' High and low risk scores could not grant entry into Checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Only if a person who is medium risk is wrongly classified as high or low risk would they be erroneously denied entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' If individuals who are low or high risk were erroneously classed as medium risk, they would then be granted entry, when in fact the policy intention was to exclude them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' On the presumption of harmlessness, see A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Ashworth and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Zedner, Preventive Justice (Oxford: OUP, 2014), 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 11 algorithms of the kind under consideration are conventionally and primarily evaluated in terms of accuracy in predicting the correct classification of candidates taken from the ‘test set’ (or ‘out of bag sample’) namely data held back and kept separate from the remainder of the dataset used to train and tune the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='54 Accuracy in this context conventionally refers to the percentage of accurate predictions calculated as a ratio of the total number of correct predictions (that is, the total number of true positives and false negatives) relative to the total number of predictions generated by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='55 But there are many other commonly used quality metrics that can be employed to evaluate a tool’s ‘accuracy’ with Krafft and Zweig56 identifying 31 different and ‘frequently used’ quality measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' These include, for example, precision and recall, which we have already considered57, as well as the AUC-ROC curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='58 They argue that developers should select quality measures that are most suited to the social and organisational context in which the algorithm is to be used, yet they observe that this is not always the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='59 Furthermore, these mathematical assessments of ‘accuracy’ typically overlook the fact that the datasets themselves are of dubious validity as a basis for predicting what an individual will do if publicly released: although they will include data on whether a person who was released was subsequently arrested, they will not contain data about whether those detained in custody would have committed a crime had they been released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' There is, however, increasing recognition by data scientists of the need to evaluate ‘quality’ from values other than accuracy in prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Those working within the field of fair-ML or ‘FAccT’ computing are actively investigating how ML models may be evaluated by reference to values such as ‘fairness,’ ‘explainability’ and ‘transparency.’60 Yet these approaches typically adopt a very narrow and somewhat contrived understanding of normative values, conceived largely in mathematical terms which then lend themselves to quantification and computational analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='61 This is particularly true of ‘fairness’, with researchers seeking to define aspects of inherently vague notions of societal fairness in mathematical terms in order to incorporate fairness ideals into machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='62 However, as critics observe, these definitions are simplifications that fail to capture the full range of similar and overlapping notions of fairness and discrimination in philosophical, legal and sociological contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='63 Apart from the dangers associated with relying upon ML-based prediction models for which a causal understanding of the variables remains unknown (examined in Part I), these quality metrics focus exclusively on the algorithmic tool rather than the larger socio-technical system in which they operate producing a ‘framing 54 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Michie, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Spiegelhalter and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Taylor (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=') Machine Learning, Neural and Statistical Classification (New Jersey: Prentice Hall, 1995), 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' A ‘validation set’ refers to at least one portion of data from the training dataset conventionally set aside to validate the model and tune its parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 55 Kaitlin Kirasich, Trace Smith and Bivin Sadler, “Random forest vs logistic regression: Binary classification for heterogenous datasets” (2018) 1(3) SMU Data Science Review 1-24, 12-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 56 Tobias D Krafft and Katharina A Zweig “So far, So Good: Multidisciplinary perspectives on algorithms, decisions and algorithmic decision-making – Computer Science Dimensions.” (2019) Unpublished manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 57 See section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 58 Area under the curve – receiver operating characteristic: Melissa Hamilton “Adventures in risk: predicting violent and sexual recidivism in sentencing law” [2015] Arizona State Law Journal 47(1) 11-62, 34-37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 59 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', demonstrating that the purported AUC-ROC score of COMPAS (an algorithmic tool developed by Northpointe Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' to undertake ‘recidivism risk’ assessment and inform sentencing decisions), offers very little meaningful information about the tool’s accuracy in, nor suitability for, this purpose: Krafft and Zweig ‘So far, So Good…’ at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 60 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', see the ACM Conference on Fairness, Accountability, and Transparency (ACM FAccT): https://facctconference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='org/;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Roth and Kearns, The Ethical Algorithm: The Science of Socially Aware Algorithm Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 61 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Barabas, “Beyond bias: ‘Ethical AI’ in criminal law” in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Dubber, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Pasquale and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Das (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' ), The Oxford Handbook of Ethics of AI (OUP, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=" 62 See the well-known debate about the fairness of COMPAS risk scoring: Julia Angwin, Jeff Larson, Surya Mattu, and Lauren Kirchner, 'Machine bias: There’s software used across the country to predict future criminals." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' And it’s biased against blacks” (ProPublica, 2016) at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='propublica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='org/article/machine-bias-risk-assessments-in-criminal- sentencing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' William Dietrich, Christina Mendoza and Tim Brennan, “COMPAS risk scales: Demonstrating accuracy equity and predictive parity (2016)” Northpointe Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' https://go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='volarisgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='com/rs/430-MBX- 989/images/ProPublica_Commentary_Final_070616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 63 Ben Green, “The false promise of risk assessments: Epistemic reform and the limits of fairness” (2020) FAT*’20: Proceedings of the 2020 Conference on Fairness, Accountability and Transparency, 594-606;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Roth and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Kearns, The Ethical Algorithm: The Science of Socially Aware Algorithm Design, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 12 trap’ that leads to inappropriate and misleading ‘quality’ guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='64 We argue that algorithmic tool ‘quality’ also demands adherence to the basic constitutional principles and legal requirements, particularly when used to inform ‘rights-critical’ decisions by public authorities constituting constitute vitally important and relevant socio-technical context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Hence the ‘quality’ of these tools should be assessed as constitutionally and legally acceptable before they can be used to inform real-world decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' These assessments require careful, context- sensitive, qualitative legal evaluation that cannot be collapsed or reduced to purely mathematical terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' To do so risks overlooking rights-critical normative choices made by developers in the design process of algorithmic tools, enhancing the dangers of illegality and injustice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='4 Step 4: Develop a digital interface for use in a particular organisational setting Armed with a suitable prediction model, a digital user-interface is then designed to communicate the model’s predictions to human decision-makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' The widespread take-up of smart devices substantially enhances the ease and convenience with which front-line officers can access algorithmic predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Although convenience, efficiency and seamlessness are understandably important from a user-design perspective, there is a worrying tendency to downplay or overlook the need to design interfaces in ways that respect the due process rights of affected persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' We have already argued that respect for due process rights should be demonstrated when configuring error thresholds within algorithmic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' This is also true when designing digital user-interfaces for front-line decision-makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Although identifying the specific set of procedural requirements (and concomitant duties) that the right to due process imposes requires is only possible in specific application contexts, three matters are especially important yet often overlooked by UX designers trained to prioritise organisational convenience and efficiency, briefly outlined below: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' The right to reasons: under British Administrative law, an individual’s right to reasons for public authority decisions that have significant adverse effects flows from the legal duty of public authorities to explain and justify their decisions in accordance with law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='65 If an algorithmic tool produces a score and/or recommendation about an individual without an accompanying explanation of what it is intended to signify, or how it was generated, the front-line official receiving it may not be able to provide the affected individual with reasons to support it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Accordingly, user-interfaces for algorithmic tools to support public authority decision-making should, as a matter of constitutional best practice, if not of legal obligation, provide functional explanations to accompany the tool’s output(s) (at least for ‘interpretable’ ML models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Front-line officers should also be able to identify and cross-check how the output was produced, lest over-simplified explanations exacerbate the likelihood of automation bias, discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='66 Similarly, tool outputs must not be mislabelled: a tool intended to predict ‘recidivism risk’ developed from arrest data should not be called a ‘recidivism’ predictor, but a ‘re- arrest’ predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' The legal duty of public authorities to give reasons for decisions that entail the adverse treatment of individuals suggests that they should not employ decision-support tools that use advanced ML techniques such as deep neural networks, for which even functional explanations remain elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='67 Yet, for the three case studies examined here, we could not identify whether front-line decision-makers were automatically provided with explanations to accompany the algorithmic predictions intended to assist them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Much more extensive and systematic transparency regarding decision-support tools used in criminal justice is therefore needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='68 ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' The right to contest: providing explanations about the tool’s functional logic, including the relevant variables upon which it relies, might help decision-makers discharge their administrative law duty to provide affected individuals with ‘reasons’ for a particular decision in specific cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' However, those reasons must be lawful reasons69 and good faith decision-making is not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Hence the individual’s right to challenge and 64 Selbst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=', “The legal construction of black boxes.” 65 R v Secretary of State for the Home Department, Ex parte Doody [1993] UKHL 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Jennifer Cobbe, “Administrative Law and the Machines of Government: Judicial Review of Automated Public-Sector Decision-Making” 39(4) Legal Studies 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 66 We are indebted to Mireille Hildebrandt for this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 67 See Rebecca Williams, “Rethinking administrative law for algorithmic decision-making” (2021) 42(2) Oxford Journal of Legal Studies 468-494, 482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 68 See section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 69 Williams “Rethinking administrative law for algorithmic decision-making,” 482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 13 contest such decisions has considerable constitutional importance, reflected in rights to liberty, to due process and to a fair trial protected under ECHR Articles 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' While the Data Protection Act 2018 (and the GDPR and Law Enforcement Directive upon which the 2018 Act builds) confers on data subjects a right to contest fully automated decisions,70 contestation rights must also accrue to those significantly adversely affected by a recommendation produced by an algorithmic tool to guard against injustice and the abuse of power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' The right to an unbiased tribunal: Although we have emphasised the ‘fair hearing’ limb of the administrative law right to procedural fairness, its second limb - the ‘rule against bias’ - is equally important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' It requires that decision-making tribunals must be ‘impartial’, meaning free from both actual bias and the appearance of bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='71 The issue of bias and discrimination, particularly racial and gender bias of algorithmic tools used in criminal justice decision-support, has received widespread attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='72 Datasets inevitably reflect underlying biases in the historic social practices to which the data pertains so that, if used to generate prediction models, the resulting outputs will reflect and reinforce these biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Historically marginalised groups are thus subjected to a higher risk of unjust discrimination relative to individuals from majority groups because the resulting outputs systematically discriminate unjustly (rather than being arbitrary, they are systematically patterned for reasons we can point to) and may violate the right to be free from unjust discrimination in the determination of opportunities and burdens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' To this end, the ICO concluded that the London Gangs Matrix was unlawful because the MPS failed to ensure that its use complied with the Public Sector Equality Duty (s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='149 of the Equality Act 2010) because it disproportionately singles out black men relative to other ethnic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='73 Less attention, however, has been paid to automation bias,74 referring to the human tendency to trust machine- made judgments over their own despite their potential or demonstrated capacity for error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Although algorithmic ‘recommender’ tools may formally preserve human judgement, front-line officials may in practice tend to follow recommendations unreflectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='75 As Johnson and Powers76 commented in the context of computerised aviation systems in which a human ‘in-the-loop’ is expected to supervise computational systems, those individuals may be understandably reluctant to intervene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='77 Similarly, front-line criminal justice decision- makers under considerable time pressures, heavy workloads and lack a clear understanding of how algorithmic recommendations are produced are unlikely to depart from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Accordingly, officials who use algorithmic tools must be properly trained so that they can properly understand and interpret algorithmic recommendations and their limitations, and to ensure they exercise meaningful independent judgement rather than unthinkingly following the tool’s outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Discussion and Recommendations for Reform We have demonstrated how important constitutional principles and legal duties are implicated at every step of the algorithmic model-building process for use by criminal justice decision-makers yet largely overlooked in relation to three such tools used to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' As a result, these tools may significantly enhance the risk that decisions based on their predictions may be unjust or may otherwise entail the unlawful exercise of decision-making authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' To minimise these dangers, it is vital that during the algorithmic tool-building process, proper consideration is given to: (a) the nature of the substantive interventions that flow the outputs generated by these prediction tools, particularly effects on the rights, interests, and legitimate expectations of those subjected to algorithmic evaluation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 70 Williams “Rethinking administrative law for algorithmic decision-making ,” 474-476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 71 Ridge v Baldwin [1964] AC 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 72 Cathy O’Neil, Weapons of Math Destruction (New York: Crown Books, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 73 Information Commissioner’s Office, “Enforcement Notice to the Commissioner of Police of the Metropolis” at [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 74 Linda J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Skitka, Kathleen L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Mosier, Mark Burdick, and Bonnie Rosenblatt, “Automation bias and errors: Are crews better than individuals?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' (2000) 10(1) The International Journal of Aviation Psychology 85-97 (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' ICO (n 101) at [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 75 Hartmann & Wenzelburger, “Uncertainty, risk and the use of algorithms in policy decisions,” 269-287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 76 Deborah G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Johnson and Thomas M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Powers, “Computer systems and responsibility: A normative look at technological complexity” (2005) 7 Ethics and Information Technology 99-107, 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 77 Arthur Kuflik, “Computers in control: Rational transfer of authority or irresponsible abdication of autonomy?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' (1999) 1 Ethics and Information Technology 173-184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 14 (b) the legal duties and obligations that apply to all public officeholders who make decisions about the treatment of individuals within the criminal justice process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' (c) the public policy objectives that the larger criminal justice system in which the tool is embedded is intended to serve, and (d) the need to maintain the general public’s trust and confidence in the integrity of the criminal justice system and the administration of justice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Yet, as we argued in Part I, computer scientists are conventionally trained to abstract or ‘detach’ the prediction model from legally and constitutionally relevant considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' By focusing exclusively on developing mathematical models to generate ‘accurate’ predictions understood in narrow, technical terms, the resulting tool may violate legal requirements and constitutional principles, resulting in decisions that entail the unlawful or improper exercise of public power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Despite the intuitive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' common-sense need to consider real-world consequences and legal concerns when approaching the task of algorithmic tool-creation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' they tend to be ignored by technical experts who may conventionally regard them as outside the relevant ‘problem space’ and hence ‘not their responsibility.’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' It is therefore essential that technical experts work collaboratively with legal professionals with a strong grasp of the appropriate constitutional principles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' human rights norms and doctrines of administrative law when creating,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' testing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' and evaluating algorithmic tools prior to their deployment by public decision-makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='78 However, because these interdisciplinary teams may want more tractable, concrete guidance to better understand how constitutional principles should apply to, and be ‘operationalised’, in the tool-building process, we offer brief advice, focusing on three sets of ‘detachment’ practices that should be steadfastly avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Detaching the tool from the substantive interventions that its predictions seek to inform We have shown that algorithmic tools cannot legitimately be designed, evaluated nor deployed without due consideration of their real-world consequences, requiring careful attention to the specific context of their deployment and application domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Yet technical specialists conventionally focus on the accuracy of algorithmic predictions, without taking due account of the effects of the substantive interventions which follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Thus, the consequences of mistaken outputs upon individuals subjected to algorithmic evaluation are not given due consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Although mistaken on-line consumer product recommendations may be inconsequential to the recipient, this is not true of mistaken, biased, or spurious algorithmic predictions that inform rights-critical criminal justice decisions, particularly about coercive detention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Throughout the tool-building process, attention must be paid to the substantive interventions intended to follow from predictions thereby generated, particularly their resulting impact on the affected individual, to ensure that: § The model’s error thresholds appropriately reflect the due process rights of the affected individual, including the right to be presumed innocence (and the presumption of harmlessness) where appropriate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' § Due consideration is given to the normative acceptability of utilising the algorithmic models to generate predictions to inform decisions that may adversely affect the individual under evaluation, particularly in the absence of scientific evidence of causal relations upon which the tool relies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Algorithmic tools should not be deployed in rights-critical contexts unless a plausible account of why the features upon which the model relies to generate productions may be expected to provide reliable and truthful indicators of the phenomenon which the model seeks to predict79;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' and § The design of the algorithmic tool’s user-interface for communicating outputs to front-line officers must enable them to understand what the output signifies—including its limitations—and remind the officeholder 78 Michael Veale, Max Van Kleek and Reuben Binns, ‘Fairness and accountability design needs for algorithmic support in high-stakes public sector decision-making’ [2018] Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems 440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 79 For further discussion, see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' of Part I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 15 of the rights of affected persons to demand reasons for any significant adverse decision, to contest the decision, to an unbiased tribunal, and to help the officer ensure that meaningful, independent judgement is brought to bear when making the resulting decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='2 Detaching the tool from its surrounding legal context and/or underlying policy purpose Failing to properly attend to the surrounding legal and policy context during the design and construction of algorithmic decision-support tools enhances the likelihood that these tools will produce predictions that give rise to unlawful decision-making and may generate serious injustice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Accordingly, when building such tools, due care must be given to: § particular matters which the law requires the decision-maker to decide, including specific considerations mandated by law that must be considered in decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' For example, if the statute requires consideration of the likelihood of an individual committing a crime in future, then it should not be assumed that this is equivalent to the likelihood of being arrested in future;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' § ensuring that only ‘legally relevant’ considerations (‘features’) are analysed in the model generation process, and that irrelevant features are disregarded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' § attending to the congruence between the phenomenon that the tool is intended to predict and the substantive decisions which the public official is legally obliged to make.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Thus, if the decision-maker is required to consider the ‘dangerousness’ of an individual, predictions about the likelihood of being arrested in future may not be reliable indicia of dangerousness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' and § careful evaluation of whether the proposed ground truth data constitutes a valid and acceptable proxy for generating predictions about the phenomenon of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='3 The need for systematic safeguards: attending to the power, opacity, and scale of algorithmic tools We have highlighted how algorithmic risk assessment tools used for criminal justice purposes may result in the unjust and/or unlawful treatment of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' But transparency and accountability must also be provided to the public at large, otherwise there is no basis for them to trust that governmental power is not being abused, exploited, or otherwise exercised corruptly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Accordingly, the exercise of governmental authority—particularly in criminal justice contexts—must be subject to systematic, institutional oversight, to ensure that this authority is exercised in a lawful and accountable manner, including mechanisms to ensure that the exercise of that authority in individual cases is subject to meaningful contestation, review, and redress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' A commitment to constitutionalism demands that endowed with governmental power exercise it in an open and transparent manner, routinely rendering an account for the exercise of their decision-making power to the community at large from whom their power is ultimately derived and on whose behalf they purport to act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Yet relatively little attention is given to the opacity and power of these tools arising from their scale and speed of operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Instead, we observe a third, equally if not more troubling form of ‘detachment’, generating serious constitutional dangers: the inapposite use of analogical reasoning evident in some policy-making and judicial reasoning, whereby advanced data-driven tools are detached from their capacity to be employed automatically and at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Instead, algorithmic tools are often treated as ‘equivalent’ to existing (analogue) tools long used to inform criminal justice decisions, on the basis that they pursue the same purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' This ‘fallacy of equivalence’ is used in at least two ways in legal analysis that are both mistaken and dangerous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='80 Firstly, it is constitutionally mistaken to assume that the legal basis authorising the use of an ‘old fashioned,’ handcrafted, statistical tool also provides the legal basis for a data-driven algorithmic tool capable of being automated at scale, merely because the latter serves the ‘equivalent’ purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' This inappropriate invocation of the argument from analogy overlooks the massively enhanced power of data-driven technologies, their opacity, the susceptibility of humans to automation bias, and the instant reproducibility, transfer and storage of digital 80 Karen Yeung, “Constitutional Principles in a Networked Digital Society” (2022), https://ssrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='com/abstract=4049141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 16 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='81 For example, in a judicial review challenge to the South Wales Police use of live facial recognition technology (FRT), the High Court reasoned that having one’s face subjected to analysis by live FRT was no different from having one’s face photographed by an police officer in the course of his or her duty, which earlier judicial decisions had confirmed may be lawfully undertaken by police on the basis of their common law powers to prevent and detect crimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='82 Such reasoning reflects a failure to understand that the technological capacities of contemporary data-driven, networked technologies are qualitatively different from an earlier generation of pre-internet enabled tools, used for the same or similar purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' The danger here is that the false assertion of conformity may undermine the rule of law, while bypassing the need for public debate and deliberation about whether the general public considers it appropriate for these technologies to be employed by law enforcement authorities, and if so, on what terms and with what safeguards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Accordingly, we argue that the capabilities and limitations of new technological tools in real-world contexts must be properly considered when scrutinising claims that existing warrants of authority provide an adequate legal basis to authorise their use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' If they expand and extend dangers associated with their use, express statutory authorisation must first be provided, facilitating open public debate and scrutiny rather than acquiesce in the expansion of the state’s coercive powers by technological stealth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Secondly, the capacity of decision-support tools to operate automatically, at scale, yet in a highly opaque manner, must be taken into consideration when evaluating the ‘proportionality’ of deploying any tool, even assuming that it serves a legitimate and lawful purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' We applaud the reasoning of the court in SyRI when concluding that, although state authorities could legitimately employ data-driven tools to identify welfare fraud, the gathering of administrative data from a wide range of unconnected sources to create highly detailed profiles of individuals was disproportionate, violating the ECHR Article 8(1) right to privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='83 In so doing, the Court provided important judicial acknowledgement that analogue tools cannot be considered ‘equivalent’ to their networked data-driven digital counterparts, even when employed to serve familiar and legitimate legal and policy objectives, owing to qualitative differences arising from their capacity to scale, and the level of intrusiveness that systematic data-collection across disparate contexts entails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Although algorithmic tools offer organisations the possibility of automating tasks at scale, this radically magnifies their power and, in turn, the scope at which injustice and the abuse of power that they may generate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Indeed, the recent UK Post Office Horizon scandal in which defective financial accounting software produced by Fujitsu resulted in the wrongful conviction of 39 innocent individuals, considered the largest miscarriage of justice in British history, powerfully illustrates how software can radically ‘scale injustice’ in a systemic yet highly opaque manner which individuals may find practically impossible to contest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='84 It highlights the urgent need for systematic transparency, accountability and independent oversight to protect those subjected to algorithmic evaluation within the criminal justice system, which our analysis reveals are sorely lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' In particular, institutional mechanisms of transparency and accountability are necessary in the decision to adopt these tools in the first place, and the way in they are built, deployed, and evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' The UK currently has no comprehensive, publicly available, inventory of decision-support tools used by criminal justice authorities that provides even basic information about such tools, let alone supplying evidence of their beneficial and adverse impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Indeed, there may be tools in use of which the public has no knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='85 Without meaningful systematic oversight, algorithmic tools will continue being built and used in ways that automate and scale injustice, enhancing the risks that governmental power will be abused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Systematic independent oversight is urgently needed, including the creation and maintenance of a transparency register to help ensure that the safeguards we have identified have been properly implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='86 Such a register should, at minimum, include 81 Skitka et al, “Automation bias and errors: Are crews better than individuals?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 82 R (Bridges) v South Wales Police and ors [2020] EWCA Civ 1058 at [54]-[85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 83 SyRI Judgment at [6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='78]-[6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 84 Haroon Siddique, “Wrongly convicted Post Office workers to get up to £100,000 interim payouts” (The Guardian, 22 July 2021), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='theguardian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='com/business/2021/jul/22/wrongly-convicted-post-office-workers-to-get-up-to- 100000-interim-payouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 85 House of Lords Justice and Home Affairs Committee, “Technology rules?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 40: “On the most basic level, we cannot be certain what technologies are being used for the application of the law in England and Wales.” 86 Also recommended by the House of Lords Justice and Home Affairs Committee, “Technology Rules?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' at pp 45-46, relating to “all advanced algorithms used in the application of the law that have direct or indirect implications for individuals.” 17 public information about the matters identified in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='2(d), as part and parcel of the ‘right to reasons’ including: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' The legal authority upon which the tool may lawfully be developed and used, including the particular decisions that the tool is intended to inform;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' The configuration of error thresholds and other normative trade-offs made during tool-development;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Information concerning how those normative choices were made and by whom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Input, test, and validation data used to produce the underlying mathematical model, and upon which the tool relies to generate predictions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Basic information about the outputs produced and what those outputs are taken to signify;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' The type of machine learning used and why;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Measures concerning the tool-validation process employed, including relevant quality measures (including accuracy, recall and sensitivity);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' The extent to which the model’s predictions are supported by evidence demonstrating the causal underpinnings of the claimed relationship between the features used and the phenomenon which the model seeks to predict (or at least reasonable plausibility between those relations);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' The intended ‘users’ of the tool, and the particular organisational and legal contexts in which they are intended to be used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' The range of specific procedural and other safeguards in place to guard against decisions taken on the basis of error, illegality, irrationality or other forms of abuse of power, both individual and systematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Conclusion This two-part paper has demonstrated how seemingly ‘technical’ choices made by developers when building algorithmic tools for criminal justice authorities have serious constitutional implications which cannot be reduced to issues of technical computational know-how.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' We have argued that, because technical developers cannot reasonably be expected to have a proper understanding of public law principles, they must collaborate closely with public law experts when deciding whether to employ decision-support tools for specific criminal justice purposes, and if justified, to ensure they are configured in a manner that is demonstrably compliant with public law principles and doctrine, including respect for human rights, throughout the tool-building process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' It is widely recognised that public law principles and safeguards apply to inform and constrain the use of weapons and other enforcement tools and techniques employed by the police and other criminal justice authorities, such as handcuffs, guns, tasers, teargas through to heightened surveillance, electronic tagging, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Yet there has been a peculiar ongoing failure to recognise that these same principles and safeguards should be applied to constrain and inform algorithmic decision-tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Although public law scholars have highlighted how algorithmic tools may be in tension with constitutional values, they have been surprisingly reluctant to argue, and to demonstrate why constitutional principles operate as ‘red lines’ that mark out the boundaries of acceptability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' In contrast, by employing cross-disciplinary insights from public law and data science, we have sought to demonstrate that unless and until these tools are designed and deployed in compliance with basic constitutional principles and legal requirements, they should not be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='87 Systematic institutional oversight mechanisms are urgently needed to ensure such compliance, otherwise algorithmic tools are likely to proliferate in ways that violate individual rights, producing injustice and eroding public trust in the integrity of the criminal justice system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content='22 12158 words 87 Tobias D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Krafft, Katharina A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' Zweig and Pascal D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} +page_content=' König, “How to regulate algorithmic decision-making: A framework of regulatory requirements for different applications” (2020) Regulation and Governance, 1-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E3T4oBgHgl3EQfyAsA/content/2301.04715v1.pdf'} diff --git a/QNFRT4oBgHgl3EQf6zgQ/content/tmp_files/2301.13677v1.pdf.txt b/QNFRT4oBgHgl3EQf6zgQ/content/tmp_files/2301.13677v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..42308895bc475b6e2dac2139f847c47177e67ca2 --- /dev/null +++ b/QNFRT4oBgHgl3EQf6zgQ/content/tmp_files/2301.13677v1.pdf.txt @@ -0,0 +1,1187 @@ +arXiv:2301.13677v1 [math.AP] 31 Jan 2023 +SOME RIGIDITY RESULTS AND ASYMPTOTIC PROPERTIES FOR +SOLUTIONS TO SEMILINEAR ELLIPTIC P.D.E. +MATTEO RIZZI AND PANAYOTIS SMYRNELIS +Abstract. We will present some rigidity results for solutions to semilinear elliptic equations of +the form ∆u = W ′(u), where W is a quite general potential with a local minimum and a local +maximum. We are particularly interested in Liouville-type theorems and symmetry results, which +generalise some known facts about the Cahn-Hilliard equation. +MSC2020: 35B06; 35B50; 35B53. Keywords: Liouville theorems, radial symmetry, rigidity, +Cahn-Hilliard equation. +1. Introduction and main results +Our aim is to revisit some asymptotic properties and rigidity results for solutions u to the +semilinear elliptic P.D.E. +(1.1) +∆u(x) = W ′(u(x)), u ∈ C2(D), D ⊂ Rn, n ≥ 2, +with a potential W ∈ C1,1 +loc (R). The domains D ⊂ Rn we consider are connected open unbounded +sets such that +(1.2) +∀R > 0, D contains a ball of radius R. +We shall also assume that on D the solution u takes its values in a bounded interval where W is +monotone. For instance, one may consider the Cahn-Hilliard equation +(1.3) +∆u = W ′(u) = u3 − u + δ +in Rn, +with |δ| < +2 +3 +√ +3, so that the polynomial f(t) = t3 − t + δ admits exactly three real roots +z1(δ) < − 1 +√ +3 < z2(δ) < +1 +√ +3 < z3(δ), +and is negative on the interval (z2(δ), z3(δ)). This equation was largely studied in the literature. +For example, some particular solutions were constructed in [15] and [17], while some results about +radial and cylindrical symmetry of solutions and Liouville type results can be found in [21]. Our +starting point is the following theorem. +Theorem 1.1. [[21]] Let n ≥ 2, δ ∈ (− +2 +3 +√ +3, +2 +3 +√ +3) and let uδ be a solution to (1.3) such that +(1.4) +uδ > z2(δ) +outside a ball BR ⊂ RN. +(1) If δ ∈ (− +2 +3 +√ +3, 0], then u ≡ z3(δ). +(2) If δ ∈ (0, +2 +3 +√ +3), then uδ is radially symmetric (not necessarily constant). +1 + +2 +MATTEO RIZZI AND PANAYOTIS SMYRNELIS +Similar results in the case δ = 0 can be found in [10, 11]. +The purpose here is to extend Theorem 1.1 to more general non linearities. +The proofs in [21] are based on some known symmetry results (see [12]) which rely on the mov- +ing planes method. A key tool in these methods is the maximum principle, even for unbounded +domains (see [1]). +If +(1.5) +u(D) ⊂ [a, b], and W ′ < 0, on [a, b) (with a, b ∈ R), +it is straightforward by Lemma 2.1 below that +(1.6) +lim +d(x,∂D)→∞ u(x) = b, and W ′(b) = 0. +Thus, we shall focus on the more involved problem where +(1.7a) +u(D) ⊂ (a, b], +(1.7b) +W ′(a) = W ′(b) = 0, and W ′ < 0, on (a, b) (with a, b ∈ R). +If we assume in addition to (1.7), the nondegeneracy condition: +(1.8) +W ′(s) ≤ −C0(s − a) on [a, s0], for some C0 > 0 and s0 ∈ (a, b), +we can apply comparison arguments of Berestycki, Caffarelli, and Nirenberg [1, Lemma 3.2] to +deduce that +(1.9) +d(x, ∂D) > η ⇒ u(x) ≥ a + ǫ, for some constants η, ǫ > 0. +Consequently, the asymptotic property (1.6) follows again from Lemma 2.1. +On the other hand, in the degenerate case where (1.8) does not hold, the asymptotic behaviour +of the solutions may be more involved. In the case where D is the complement of a ball, we can +relax condition (1.8) by assuming that +(1.10) +W ′(s) ≤ −C0(s − a)p on [a, s0], for some C0 > 0, p < +n +n − 2, and s0 ∈ (a, b), +Under assumption (1.10), we can still prove the asymptotic property (1.6) for solutions provided +(1.7) holds (cf. Proposition 2.2). However, in the case of potentials such that +lim +u→a+ +W ′(u) +(u − a)p = −λ +for some λ > 0 and p > +n +n−2, radial solutions u : Rn → (a, b) of (1.1) satisfying +(1.11) +lim +|x|→∞ u0(x) = a +may exist in dimensions n ≥ 3 (cf. Lemma 2.3). Therefore, condition (1.10) is optimal to derive +(1.6), when D is the complement of a ball. For general domains, condition (1.10) is not sufficient +to deduce the asymptotic behaviour of the solution. In Proposition 2.6, we construct a solution of +(1.1) in a dumbbell shaped domain D ⊂ R2, such that u ≈ a on the one side of the neck, while +u ≈ b on the other side. +To sum up these results, we now state + +SOME RIGIDITY RESULTS +3 +Theorem 1.2. Let W ∈ C1,1 +loc (R) be a potential satisfying (1.7b). +(i) Assume u ∈ C2(Rn) is a solution of (1.1) such that u(Rn) ⊂ [a, b]. Then, when n = 2, or +n ≥ 3 and (1.10) holds, we have either u ≡ a, or u ≡ b. Otherwise (when n ≥ 3 and (1.10) +does not hold), we have either u ≡ a, or u ≡ b, or1 +(1.12) +� +u(Rn) ⊂ (a, CW ], for a constant CW ∈ (a, b) depending only on W, +lim inf|x|→∞ u(x) = a. +(ii) Assume the domain D satisfies (1.2), and u ∈ C2(D) is a solution of (1.1) such that +u(D) ⊂ (a, b]. Then, we have limd(x,∂D)→∞ u(x) = b, provided that (1.8) holds. +Next, we derive some Liouville type results by considering domains D ⊂ Rn satisfying the +following condition: +(1.13) +the radii of the balls contained in Rn \ D are uniformly bounded by a constant Λ > 0. +Theorem 1.3. Let W ∈ C1,1 +loc(R) be a non negative potential satisfying (1.7b), and W(b) = 0. +Assume the domain D satisfies (1.13), and u ∈ C2(Rn) is a bounded entire solution of (1.1) such +that supRn u = b, and u(D) ⊂ (a, b]. Then, u ≡ b. +Remark 1.4. Modica [19] proved that if W ∈ C2(R) is a non negative potential, and u is a +bounded solution of (1.1) in Rn, then the condition W(u(x0)) = 0 for some x0 ∈ Rn implies that u +is constant. In the sequel, a new proof of this result which also applies to potentials W ∈ C1,1 +loc(R) +was proposed in [4]. Therefore, the hypothesis supRn u = b in Theorem 1.3 is not very strong, since +the condition u(D) ⊂ (a, b] yields that either u < b in Rn or u ≡ b, so that supRn u ≤ b. +Since the linear behaviour of W ′ near the local maximum (see condition (1.8)) implies that +limd(x,∂D)→∞ u(x) = b, when D satisfies (1.2) (cf. Theorem 1.2 (ii)), we obtain a first corollary of +Theorem 1.3: +Corollary 1.5. Let W ∈ C1,1 +loc(R) be a non negative potential satisfying (1.7b), (1.8) and W(b) = 0. +Assume the domain D satisfies (1.2) as well as (1.13), and u ∈ C2(Rn) is a bounded entire solution +of (1.1) such that u(D) ⊂ (a, b]. Then, u ≡ b. +Finally, we particularise Corollary 1.5 in the case where D is the complement of a ball. +Corollary 1.6. Let W ∈ C1,1 +loc (R) be a nonnegative potential satisfying (1.7b), and W(b) = 0. +Assume u ∈ C2(Rn) is an entire solution of (1.1) such that +(1.14) +u(x) ∈ (a, b] +∀ x ∈ Rn\BR, +for some R > 0. Then, u ≡ b, provided that n = 2, or n ≥ 3 and (1.10) holds. +We will prove these results in Section 2. +Remark 1.7. Corollary 1.6 was established in [21] for entire solutions to the Cahn-Hilliard equation +(1.3). Here we extend this result to general nonlinearities under optimal assumptions. Indeed, the +necessity of condition (1.10) (when n ≥ 3) for Corollary 1.6 to hold, is clear in view of Lemma 2.3. +1For instance, let u0 be the radial solution provided by Lemma 2.3. +Then, by taking u(x1, . . . , xn, xn+1) = +u0(x1, . . . , xn), we can see that (1.12) holds. + +4 +MATTEO RIZZI AND PANAYOTIS SMYRNELIS +Other Liouville type results for stable solutions to semilinear PDEs were established in [8]. Here +there is no stability assumption. +After that, we will address the issue of radial symmetry. +In [13], the authors prove radial +symmetry of solutions to fully nonlinear equations of very general form, provided these solutions +have a suitable asymptotic polynomial decay at infinity (see Theorem 4 there). Here we are in- +terested in radial symmetry of solutions to (1.1) with W satisfying (1.7b), assuming that either +lim|x|→∞ u(x) = b or lim|x|→∞ u(x) = a. +The case in which lim|x|→∞ u(x) = b is easier. The following result is a consequence of [13, +Proposition 1]: +Proposition 1.8. Let u : Rn → R be a solution to ∆u = W ′(u), n ≥ 3, where W ∈ C2(Rn) is a +potential fulfilling (1.7b) and such that +(1.15) +[b − δ, b] ∋ t �→ W ′(t) +|t − b|p is H¨older continuous for some δ > 0, p ≥ n + 2 +n − 2. +Assume that u < b in Rn and lim|x|→∞ u(x) = b. Then u is radially symmetric. +On the other hand, if W is convex in an interval (b−δ, b), then the symmetry result follows from +[12, Theorem 2] in any dimension n ≥ 2. Assumption (1.15) is not required anymore. In view of +Proposition 1.8 and [12, Theorem 2], we obtain the following generalisation of Theorem 1.1 (ii): +Theorem 1.9. Let W ∈ C2(R) be a potential such that W ′(t) < 0 for any t ∈ (a, b), W ′(a) = 0, +and W(t) ≥ W(b) for any t > b. In addition, we suppose that one of the following is true: +(i) n ≥ 3, and W ∈ C6,α(b − δ, b + δ), for some δ > 0 and α ∈ (0, 1). +(ii) n ≥ 2, and W is convex in (b − δ, b), for some δ > 0. +Assume also that u : Rn → R is a solution to ∆u = W ′(u) such that u(Rn\BR) ⊂ (a, b) and +lim|x|→∞ u(x) = b. Then u < b in Rn and it is radially symmetric. +Remark 1.10. +(1) If a potential W satisfies the assumptions of Theorem 1.9, then it has a +local minimum at t = b, so that W ′(b) = 0. However, this minimum is not required to be a +global one. +(2) If W ′(t) > 0 for t > b, it follows from the maximum principle that any bounded solutions u +of (1.1) in Rn, satisfies the bound u ≤ b. +(3) Let u be a solution of (1.1) in Rn. Then, if n = 2 or n ≥ 3 and (1.10) holds, the condition +u(Rn \ BR) ⊂ (a, b), implies that lim|x|→∞ u(x) = b in view of Lemmas 5.4 and 2.1 (resp. +Proposition 2.2). +(4) For the existence of radial solutions satisfying lim|x|→∞ u(x) = b, we refer to [2, Theorem +1, Theorem 4] and [16, Theorem 1.3]. +We will prove these symmetry results in Section 3. +Assuming again that W satisfies (1.7b), the description of entire solutions to (1.1) converging to +a at infinity is a much more difficult task. In that case, only a few symmetry results are available, +under somewhat restrictive hypotheses on the solution and the nonlinearity. Some results can be +found in [5], where a monotonicity assumption is required. As a particular case, their results apply +to bounded solutions to the Lane-Emden equation +−∆u = |u|p−1u + +SOME RIGIDITY RESULTS +5 +in Rn, for which several Liuoville type results are known (see for example [3, 9, 18]). For future +purposes, the main difficulty is to remove the monotonicity and convexity assumption about the +non-linearity. +By Proposition 2.2, we know that non trivial solutions can exist only if condition (1.10) is +violated. However, the fact that +(1.16) +u(Rn\BR) ⊂ (a, b) +cannot guarantee a Liouville type result (cf. +Lemma 2.3), or even radial symmetry under the +assumption that +(1.17) +lim +|x|→∞ u(x) = a. +In section 4, we check that the solutions constructed in [6], provide examples of nonradial solutions +to (1.1), such that u(x) − a changes sign in a compact set, and (1.16) as well as (1.17) hold. It +would be interesting to see if a nonradial solution satisfying u(Rn) ⊂ (a, b) and lim|x|→∞ u(x) = a, +may also exist for a potential W having a negative derivative on the range of u. To the best of our +knowledge, this is a difficult open problem. +2. Asymptotic behaviour and Liouville type results +We first prove a basic lemma on the asymptotic behaviour of solutions satisfying (1.5). +Lemma 2.1. Let D ⊂ Rn be a domain satisfying (1.2), and let u be a solution of (1.1) (W ∈ +C1,α +loc (R), α ∈ (0, 1)). Assume also that u(D) ⊂ [a, b], and W ′ < 0 on the interval [a, b) (with +a, b ∈ R). Then, limd(x,∂D)→∞ u(x) = b, and W ′(b) = 0 hold. If in addition D = Rn, then we have +u ≡ b. +Proof. We first recall that for fixed R > 0, the solution u is uniformly bounded in C2,α (for some α ∈ +(0, 1)) on the balls BR(x) satisfying d(x, ∂D) > R + 1 with x ∈ D. Let l := lim infd(x,∂D)→∞ u(x), +and let {xk} ⊂ D be a sequence such that limk→∞ d(xk, ∂D) = ∞, and limk→∞ u(xk) = l. We +set vk(y) = u(xk + y). In view of the previous estimates, we can apply the Ascoli theorem via +a diagonal argument to the sequence {vk}, and deduce that up to subsequence, vk converges in +C2 +loc(Rn) to an entire solution v∞ of (1.1). Moreover, we have +v∞(0) = l = min +y∈Rn v∞(y), +and +0 ≤ ∆v∞(0) = W ′(l) ≤ 0, +so that, l = b, W ′(b) = 0, and v∞ ≡ b. This proves that limd(x,∂D)→∞ u(x) = b, and W ′(b) = 0 hold. +In the particular case where D = Rn, we have u ≡ b, since otherwise u would attain its minimum +at a point x0 where 0 ≤ ∆u(x0) = W ′(u(x0)) < 0, which is a contradiction. +□ +Next, given a potential satisfying (1.7b), we study the existence of solutions such that u(Rn) ⊂ +(a, b), for n ≥ 3. The answer to this question depends on the growth of W ′ in a right neighbourhood +of a. In Proposition 2.2 below, we first examine the case of potentials for which (1.10) holds. +Proposition 2.2. Let n ≥ 3, let Bρ ⊂ Rn be the open ball of radius ρ centred at the origin, and +let W ∈ C1,1 +loc(R) be a potential fulfilling (1.7b), and (1.10). Then, every solution u ∈ C2(Rn \ Bρ) +to (1.1) such that u(Rn \ Bρ) ⊂ (a, b), satisfies lim|x|→∞ u(x) = b. + +6 +MATTEO RIZZI AND PANAYOTIS SMYRNELIS +Proof. Without loss of generality, we may assume that a = 0. Assume by contradiction that +(2.18) +u(Rn \ Bρ) ⊂ (0, b − η], for some η > 0 small. +Then, we have +(2.19) +W ′(u) ≤ −c1up, ∀u ∈ [0, b − η] +for a constant 0 < c1 < C0. We first examine the case where u is radial, that is, u(x) = v(|x|). As +a consequence, v solves +(2.20) +v′′(r) + n − 1 +r +v′(r) = W ′(v(r)), ∀r ∈ [ρ, ∞). +Our claim is that v′(ρ0) ≤ 0 holds for some ρ0 ≥ ρ. Indeed, otherwise, we would have +∀r ∈ [ρ, ∞) : v′(r) > 0, and v′′(r) ≤ κ := +max +[v(ρ),b−η] W ′ < 0, +which is impossible. So far, we have proved that v′(ρ0) ≤ 0 for some ρ0 ≥ ρ. By noticing that +v′(ρ0) = 0 ⇒ v′′(ρ0) < 0 in view of (2.20), one can see that v′ < 0 holds on an interval (ρ0, ρ0 + ǫ), +for small ǫ > 0. Let l := sup{r > ρ0 : v′ < 0 on (ρ0, r)}. It is clear that l = ∞, since otherwise we +would deduce that v′(l) = 0 and v′′(l) < 0, which is a contradiction. This establishes that v′ < 0 +on (ρ0, ∞). Now, it follows from (2.20) that +∀r > ρ1 : rn−1v′(r) ≤ rn−1v′(r) − ρn−1 +0 +v′(ρ0) += +� r +ρ0 +sn−1W ′(v(s))ds ≤ −c1vp(r) +� r +ρ0 +sn−1ds +≤ −kvp(r)rn, +for a constant k > 0, and for ρ1 > ρ0 large enough. Next, an integration of the previous inequality +gives +∀r > ρ1 : v1−p(r) ≥ v1−p(r) − v1−p(ρ1) +≥ k(p − 1) +2 +(r2 − ρ2 +1), +from which we deduce that v(r) ≤ ˜kr− +2 +p−1 , for a constant ˜k > 0, and for r > ρ2 > ρ1 large enough. +Since p < +n +n−2 ⇔ +2 +p−1 > n − 2, this contradicts the lower bound provided by Lemma 5.3. Therefore +the existence of a radial solution satisfying (2.18) is ruled out. +To complete the proof of Proposition 2.2, we also have to exclude the existence of non radial +solutions. Assume by contradiction that u ∈ C2(Rn \ Bρ) is a solution of (1.1) satisfying (2.18). In +view of Lemma 5.3, u satisfies the lower bound +(2.21) +u(x) > φ∗(x) = c|x|2−n, c > 0, +where φ∗ is a subsolution of (1.1), that is, ∆φ∗ = 0 ≥ W ′(φ∗). Starting from u, we shall construct +a radial supersolution φ∗ of (1.1), such that φ∗ ≤ φ∗. Let ρi,m be the rotation of angle +π +2m around +the xi coordinate axis of Rn (m ≥ 1, i = 1, . . . , n − 1), and let +Gm := {ρk1 +1,m ◦ . . . ◦ ρkn−1 +n−1,m : 0 ≤ ki ≤ 2m+1 − 1, i = 1, . . . , n − 1}. +Using spherical coordinates, one can see that given |x0| ≥ ρ, the set ∪m≥1Gmx0 is dense in the +sphere {x ∈ Rn : |x| = |x0|}. In particular, we have +(2.22) +lim +m→∞ min +g∈Gm u(gx0) = +min +|x|=|x0| u(x). + +SOME RIGIDITY RESULTS +7 +Next, we notice that for every g ∈ Gm, x �→ u(gx) solves (1.1). On the other hand, in view of the +Kato inequality, φm(x) := ming∈Gm u(gx) is a supersolution of (1.1), satisfying φ∗ ≤ φm ≤ u. In +addition, it follows from (2.22) that φ∗(x) := limm→∞ φm(x) = min{u(y) : |y| = |x|}. Finally, since +|∇φm| is uniformly bounded on Rn\Bρ, we obtain that (up to subsequence) φm conververges weakly +to φ∗ in W 1,2(BR \ Bρ), for every R > ρ. This implies that φ∗ (which belongs to W 1,2(BR \ Bρ), +for every R > ρ) is a radial supersolution of (1.1) satisfying φ∗ ≤ φ∗ ≤ u. To conclude, we deduce +from the method of sub- and supersolutions (cf. Section 5.1, and for instance [7, Lemma 1.1.1]), +the existence of a radial solution v ∈ C2(Rn \ Bρ), satisfying 0 < φ∗ ≤ v ≤ φ∗ ≤ b − η. In view of +the first part of the proof, this is a contradiction. +So far, we have established that every solution u ∈ C2(Rn \ Bρ) to (1.1) such that u(Rn \ Bρ) ⊂ +(0, b), satisfies supRn\Bρ u = b. That is, +(2.23) +∃{xk}k∈N : +lim +k→∞ |xk| = ∞, and +lim +k→∞ u(xk) = b. +Setting vk(y) := u(xk + y), and proceeding as in Lemma 2.1, we obtain that (up to subsequence) +vk converges in C2 +loc(Rn) to an entire solution v∞ of (1.1). Furthermore, since v∞(0) = b, the +maximum principle implies that v∞ ≡ b. At this stage we consider a minimizer φR ∈ H1(BR(0)) +of the energy functional +(2.24a) +˜E(v) = +� +BR(0) +�1 +2|∇v(x)|2 + ˜W(v(x)) +� +dx, +in H1 +0(BR(0)), where +(2.24b) +˜W(v) = + + + + + +W(a) +for v ≤ 0 +W(v) +for 0 ≤ v ≤ b +W(b) +for v ≥ b. +It is known that φR is a smooth radial solution of (1.1) in BR(0), such that 0 ≤ φR ≤ maxBR(0) φR := +b − δR on BR(0), for some δR > 0. In addition, we have limR→∞ δR = 0. Thus, given ǫ > 0, we can +ensure that +• δR < ǫ for some R > 0 large enough, +• and φR ≤ b − δR ≤ vk holds on BR(0), for k ≥ kR large enough. +Finally, by applying the sliding method of Berestycki, Caffarelli, and Nirenberg [1, Lemma 3.1], +we deduce that u(x) ≥ φR(0) ≥ b − ǫ, provided that |x| > ρ + R. This completes the proof of +Proposition 2.2 +□ +In the subcritical case where W ′(u) ∼ −λ|u − a|p near a, with λ > 0 and p ∈ ( +n +n−2, n+2 +n−2), we +shall see in Lemmas 2.3 and 2.4 below, that depending on the potential, there may or may not exist +a radial solution such that u(Rn) ⊂ (a, b). +Lemma 2.3. Given any n ≥ 3, p > +n +n−2 and λ > 0, there exists a potential W ∈ C2(R) fulfilling +(1.7b), and a solution u ∈ C∞(Rn) to (1.1), such that +a) limu→a+ W ′(u) +|u−a|p = −λ, +b) u is radial and radially decreasing (i.e. u(x) = ˜u(|x|), for a smooth decreasing function +˜u : [0, ∞) → (a, b)), +c) u(Rn) ⊂ (a, b), and lim|x|→∞ u(x) = a, +d) W ′′(u(0)) > 0. + +8 +MATTEO RIZZI AND PANAYOTIS SMYRNELIS +Proof. Without loss of generality, we may assume that a = 0. First, we note that the function +v(x) = +� 2((n−2)p−n) +λ(p−1)2 +� +1 +p−1 |x|− +2 +p−1 solves the equation +∆v = −λvp +in Rn\{0}. +Next, in order to eliminate the singularity at the origin, we take a smooth cutoff function ξ : R → +[0, 1] such that + + + + + +ξ = 1 in [3, ∞), +0 < ξ < 1 and ξ′ > 0 in (2, 3), +ξ = 0 in (−∞, 2], +and we consider a function ˜u : (1, ∞) → R such that +� +˜u′′(r) = ξ(r)˜v′′(r) +∀ r ∈ [1, ∞), +˜u(r) = ˜v(r) +∀r ≥ 3. +where v(x) =: ˜v(|x|). One can see that +(2.25) +˜u′′ + n − 1 +r +˜u′ < 0 +in [1, ∞). +The latter inequality is clear if r ≥ 3. In order to prove that (2.25) holds in [1, 3) too, we note that +(2.26) +˜u′(r) = − +� ∞ +r +˜u′′(t)dt = − +� ∞ +r +ξ(t)˜v′′(t)dt += ξ(r)˜v′(r) + +� ∞ +r +ξ′(t)˜v′(t)dt < ξ(r)˜v′(r) ≤ 0, +∀ r ∈ [1, 3), +so that +˜u′′ + n − 1 +r +˜u′ < ξ +� +˜v′′ + n − 1 +r +˜v′� +≤ 0, +∀ r ∈ [1, 3). +Now, we extend ˜u to a smooth even positive function on the whole R, still denoted by ˜u, fulfilling +˜u′ < 0 in (0, ∞), ˜u′′ < 0 in [0, 1), so that ˜u′′ + n−1 +r ˜u′ < 0 holds in [0, ∞), ˜u′′′(0) = 0 and ˜u(4)(0) < 0. +This can easily be done if we recall that ˜u is affine and decreasing on [1, 2]. Since ˜u is monotone +in [0, ∞), then it is invertible in this interval with inverse function β : (0, ˜u(0)] → [0, ∞). Finally, +setting +ϕ(r) := ˜u′′(r) + n − 1 +r +˜u′(r), +∀ r > 0, +and H(s) := ϕ(β(s)), for s ∈ (0, ˜u(0)], one can see that u(x) := ˜u(|x|) satisfies the equation +∆u = H(u) in Rn. We also notice that H(˜u(0)) = n˜u′′(0) < 0 and H′(˜u(0)) = (n+2)˜u(4)(0) +3˜u′′(0) +> 0. +Thus, one can find a C1 extension of H to the whole R, still denoted by H, such that H < 0 in +(0, b), for some b > ˜u(0), and H(b) = 0. By construction, we have H(u) = −λup in (0, ˜u(3)), so +that H(0) = H′(0) = 0. In order to conclude the proof it is enough to define W to be the primitive +of H. +□ +Lemma 2.4. Given any n ≥ 3, p ∈ ( +n +n−2, n+2 +n−2), and λ > 0, there exists a potential W ∈ C2(R) +fulfilling (1.7b) and limu→a+ W ′(u) +|u−a|p = −λ, for which there are no radial solutions u ∈ C2(Rn) of +(1.1) such that u(Rn) ⊂ (a, b). + +SOME RIGIDITY RESULTS +9 +Proof. Without loss of generality, we may assume that a = 0. We consider the function H(u) = +−λup on an interval [0, β], and since p ∈ ( +n +n−2, n+2 +n−2), we set ǫ = +n +p+1 − n−2 +2 +> 0. One can find a C1 +extension of H to the whole R, still denoted by H, such that +• H < 0 in (0, b), and H(b) = 0, for some b > β. Let b = κβ, with κ > 1. +• H([0, b]) = [−λµβp, 0] for some µ > 1, such that κµ < 1 + +2ǫ +n−2. +Next, define W ∈ C2(R) to be the primitive of H vanishing at 0. We claim that +(2.27) +n − 2 +2 +W ′(u)u − nW(u) > 0 on (0, b]. +Indeed, we have n−2 +2 W ′(u)u − nW(u) = ǫλup+1 on [0, β]. On the other hand, if u ∈ [β, b], then it +follows that n−2 +2 W ′(u)u − nW(u) ≥ n−2 +2 W ′(u)u − nW(β) ≥ ( +n +p+1 − n−2 +2 κµ)λβp+1 > 0. Now that +(2.27) is established, we consider a radial solution u ∈ C2(Rn) of (1.1) such that u(Rn) ⊂ (0, b). +Setting v(|x|) = u(x) and proceeding as in the proof of Proposition 2.2, one can see that v satisfies +the standard estimates v(r) = O(r− +2 +p−1 ), v′(r) = O(r− p+1 +p−1 ), and W(v(r)) = O(r− 2(p+1) +p−1 ). +To +conclude we use the well-known Pohozaev identity: +(2.28) +� r +0 +sn−1(n − 2 +2 +W ′(v(s))v(s)−nW(v(s)) +� +ds = n − 2 +2 +rn−1v(r)v′(r)+rn�|v′(r)|2 +2 +−W(v(r)) +� +. +We notice that since p ∈ ( +n +n−2, n+2 +n−2), the right hand side of (2.28) goes to 0, as r → ∞. On the +other hand, the left hand side of (2.28) is strictly positive in view of (2.27). This rules out the +existence of radial solutions such that u(Rn) ⊂ (0, b). +□ +The next Proposition examines the existence of radial solutions in the different regimes. +Proposition 2.5. Let n ≥ 3, and let W ∈ C1,1 +loc(R) be a potential satisfying (1.7b). +(i) If (1.10) holds, there are no radial solutions u ∈ C2(Rn) of (1.1) such that u(Rn) ⊂ (a, b). +(ii) If lim supu→a+ +|W ′(u)| +|u−a| +n+2 +n−2 = 0 holds, there exists a radial solution u ∈ C2(Rn) of (1.1) such +that u(Rn) ⊂ (a, b). +(iii) Otherwise, if neither (1.10) nor lim supu→a+ +|W ′(u)| +|u−a| +n+2 +n−2 = 0 hold, depending on W, there +may or may not exist a radial solution u ∈ C2(Rn) of (1.1) such that u(Rn) ⊂ (a, b). +Proof. (i) A radial solution u ∈ C2(Rn) of (1.1) such that u(Rn) ⊂ (a, b), decays to a, as |x| → ∞. +In view of Proposition 2.2, it is clear that such a solution does not exist when (1.10) holds. +(ii) Now, assume that lim supu→a+ +|W ′(u)| +|u−a| +n+2 +n−2 = 0 holds, and define +(2.29) +˜ +W(v) = +� +W(v) +for v ≤ b +W(b) +for v ≥ b. +Theorem 4 of [2] provides the existence of a radial solution u ∈ C2(Rn) of ∆u = ˜W ′(u), such that +u > a, and lim|x|→∞ u(x) = a. By the maximum principle, we have u(Rn) ⊂ (a, b), and thus u +solves ∆u = W ′(u). +Finally, (iii) follows from Lemmas 2.3 and 2.4. +□ + +10 +MATTEO RIZZI AND PANAYOTIS SMYRNELIS +As we mentioned in the Introduction, for general domains, condition (1.10) is not sufficient to +derive the asymptotic property (1.6) of solutions. +Proposition 2.6 below, provides examples of +solutions having a different asymptotic behaviour. +Proposition 2.6. Let p > 1, and let W ∈ C1,1 +loc(R) be a potential fulfilling (1.7b), as well as +(2.30) +∀u ∈ [a, b] : W ′(u) ≥ −c(u − a)p, for a constant c > 0. +Let D = {x ∈ R2 : |x2| < ψ(x1)}, where ψ ∈ C∞(R) is a positive function such that ψ(s) = λ|s|, for +|s| > ǫ (with λ , ǫ > 0 sufficiently small, depending on W). Then, there exists a solution u ∈ C2(D) +of (1.1) such that u(D) ⊂ (a, b), and +(2.31) +lim +x1→+∞ u(x) = a and +lim +x1→−∞ u(x) = b. +Proof. Without loss of generality we may assume that a = 0. We shall first construct a supersolution +φ∗ of (1.1) in D. We define the auxilliary functions +(2.32a) +f(reiθ) = r− +2 +p−1 g(θ), +with g : [−θ0, θ0] → (0, ∞) (θ0 < π +2 ), a positive solution of the O.D.E.: +(2.32b) +g′′(θ) = −cgp(θ) − +4 +(p − 1)2 g(θ). +Next, setting λ = tan(θ0), one can check that +(2.33) +∆f(x) = −c(f(x))p in the sector S = {x1 > 0, |x2| < λx1}. +In addition, we have f(x) > b in the set {0 < x1 ≤ ǫ, |x2| < λx1}, provided that ǫ > 0 is sufficiently +small. Finally, we take +(2.34) +φ∗(x) = +� +min(f(x), b) +when x1 > ǫ, and |x2| < λx1. +b +when x1 ≤ ǫ, and |x2| < ψ(x1). +Using the Kato inequality, one can see that φ∗ is a supersolution of (1.1) in D. Indeed, in view of +(2.30) and (2.33), we have +(2.35) +∆φ∗ ≤ −cf pχ{f 0 in (b, ∞) and considering the phase plane for the ODE +v′′ = ˜W ′(v). The situation is analogue to the one we have for the classical double well potential +1 +4(1 − t2)2. +It follows again from the Kato inequality that ∆φ∗ ≥ W ′(φ∗) holds in H1 +loc(D). In addition, it +is clear that φ∗ < φ∗ holds in D. Therefore, we deduce from the method of sub- and supersolutions + +SOME RIGIDITY RESULTS +11 +(cf. Section 5.1, and for instance [7, Lemma 1.1.1]), the existence of a solution u ∈ C2(D) of (1.1) +satisfying φ∗ ≤ u ≤ φ∗. Since 0 < u < b by the maximum principle, the solution u has all the +desired properties. +□ +Now, we are ready to prove Theorems 1.2 and 1.3, and their corollaries. +Proof of Theorem 1.2. (i) Assume u ∈ C2(Rn) is an entire solution of (1.1) such that u(Rn) ⊂ [a, b]. +When n = 2, u is a bounded superharmonic function defined on R2. Thus, u is constant and equal +to a critical point of W. That is, u ≡ a or u ≡ b. +In higher dimensions n ≥ 3, we have by the maximum principle either u ≡ a, or a < u ≤ b on +Rn. We shall first assume that a < u ≤ b as well as (1.10) hold, and we shall prove that u ≡ b. In +view of (1.10), Proposition 2.2 implies that lim|x|→∞ u(x) = b, and a + ǫ ≤ u ≤ b holds on Rn, for +some ǫ > 0. Thus, u ≡ b, by Lemma 2.1. +Next, we consider again an entire solution u of (1.1) satisfying u(Rn) ⊂ [a, b] in dimensions +n ≥ 3, but without assuming (1.10). By the maximum principle, we have either u ≡ a, or u ≡ b, or +a < u < b. Let +F = {u is a solution of (1.1) such that u(Rn) ⊂ (a, b)}, +CW = sup{u(x) : x ∈ Rn, u ∈ F}. +Our first claim is that +(2.38) +CW < b. +Indeed, assume by contradiction that there exists a sequence {uk} ⊂ F, and a sequence {xk} ⊂ Rn, +such that limk→∞ uk(xk) = b. Setting vk(y) = uk(xk + y), and proceeding as in Lemma 2.1, we +obtain that (up to subsequence) vk converges in C2 +loc(Rn) to an entire solution v∞ of (1.1). Fur- +thermore, since v∞(0) = b, the maximum principle implies that v∞ ≡ b. At this stage we consider +the minimizer φR ∈ H1(BR(0)) defined in (2.24). It is known that φR is a smooth radial solution +of (1.1) in BR(0), such that a ≤ φR ≤ b − δR on BR(0), for some δR > 0. In addition, by taking +R > R0 large enough, we have a < φR ≤ b − δR on BR(0). Thus, for fixed R > R0, we can ensure +that a < φR ≤ b − δR ≤ vk holds on BR(0), provided that k ≥ kR is large enough. Finally, by +applying the sliding method of Berestycki, Caffarelli, and Nirenberg [1, Lemma 3.1], we deduce that +for k ≥ kR, vk as well as uk are entire solutions of (1.1) satisfying respectively vk(Rn) ⊂ [a + ǫR, b], +and uk(Rn) ⊂ [a+ǫR, b], with ǫR := φR(0)−a > 0. In view of Lemma 2.1, this implies that uk ≡ b, +for k ≥ kR, which is a contradiction. This proves (2.38). +The fact that lim inf|x|→∞ u(x) = a holds for every u ∈ F also follows from Lemma 2.1. Indeed, +assuming by contradiction that lim inf|x|→∞ u(x) > a we would obtain that u(Rn) ⊂ [a + ǫ, b], for +some ǫ > 0. Therefore, using Lemma 2.1, we conclude that u ≡ b, which is a contradiction. +(ii) Now, assume the domain D satisfies (1.2), and u ∈ C2(D) is a solution of (1.1) such +that u(D) ⊂ (a, b]. +In the nondegenerate case where (1.8) holds, [1, Lemma 3.2] implies that +a + ǫ < u(x) ≤ b holds for some ǫ > 0, provided that d(x, ∂D) > η, for some η > 0. Thus, in view +of Lemma 2.1, we have limd(x,∂D)→∞ u(x) = b. +□ + +12 +MATTEO RIZZI AND PANAYOTIS SMYRNELIS +Proof of Theorem 1.3. On the one hand, since +sup +Rn u = b, +let {xk}k∈N ⊂ Rn be a sequence such that limk→∞ u(xk) = b, and set vk(y) = u(xk +y). Proceeding +as in the proof of Theorem 1.2, one can see that (up to subsequence), vk converges in C2 +loc(Rn) to +an entire solution v∞ ≡ b. In particular, given R > 0 and δ > 0, we have u(x) ∈ [b − δ, b], provided +that x ∈ BR(xk), and k ≥ K(R, δ) is large enough. +On the other hand, let ι := infRn u ≤ b, and assume by contradiction that ι < b and W(ι) > 0. +Next, define the auxiliary potential +˜W(u) = +� +W(u) +for u ≥ ι +W(ι) +for u ≤ ι, +and consider a minimiser φR ∈ H1(BR(0)) of the energy functional +˜E(v) = +� +BR(0) +�1 +2|∇v(x)|2 + ˜W(v(x)) +� +dx, +in the class A = {v ∈ H1(BR(0)), v = ι on ∂BR(0)}. Setting σ := min{t ≥ ι : W(t) = 0}, and +σR := supBR(0) φR, one can see that +ι ≤ φR ≤ σR < σ +holds for every R > 0, since W(ι) > 0. In addition, φR is a smooth radial solution of (1.1) in BR(0), +such that limR→∞ σR = σ. Thus by taking R > 0 large enough, we can ensure that ι < σR < b. +As a consequence, we also have φR(y) ≤ u(y + xk), provided that y ∈ BR(0), and k ≥ K(R, δ). +Finally, by applying the sliding method of Berestycki, Caffarelli, and Nirenberg [1, Lemma 3.1], we +deduce that for u ≥ σR > ι, holds on Rn, which is a contradiction. +So far we have established that W(ι) = 0, so that ι = σ. To complete the proof of Theorem 1.3, +it remains to show that ι = b. Indeed, if ι < b and W(ι) = 0, then in particular ι < a, since W ′ < 0 +on (a, b). Let {zk}k∈N ⊂ Rn be a sequence such that limk→∞ u(zk) = ι, and set wk(y) = u(zk + y). +Proceeding as previously, we obtain that (up to subsequence), wk converges in C2 +loc(Rn) to an entire +solution w∞ ≡ ι. In particular, given R0 = Λ + 1 (cf. (1.13)) and η > 0 such that ι + η < a, we +have u(x) ∈ [ι, ι + η], provided that x ∈ BR0(zk) ∩ D, and k ≥ ˜K(η) is large enough (we note that, +in view of (1.13), BR0(zk) ∩ D ̸= ∅). This is a contradiction. Therefore, we have proved that ι = b, +and u ≡ b. +□ +Proof of Corollary 1.5. Under the assumptions of Corollary 1.5, we can apply Theorem 1.2 (ii) to +deduce that limd(x,∂D)→∞ u(x) = b. On the other hand, in view of Remark 1.4 we have u ≤ b, so +that supRn u = b. Therefore, Theorem 1.3 implies that u ≡ b. +□ +Proof of Corollary 1.6. When n ≥ 3, we first apply Proposition 2.2 to deduce that lim|x|→∞ u(x) = +b. Next, in view of Remark 1.4 we obtain that supRn u = b. Finally, Theorem 1.3 implies that +u ≡ b. On the other hand, when n = 2, u is superharmonic in {x ∈ R2 : |x| > R}. Setting +γ := min|x|=R+1 u(x) ∈ (a, b), we deduce from Lemma 5.4, that u(x) ∈ [γ, b), provided that +|x| > R + 1. In view of Lemma 2.1, Remark 1.4 and Theorem 1.3, we conclude as previously that +u ≡ b. +□ + +SOME RIGIDITY RESULTS +13 +3. Radial symmetry for solutions converging to the local minimum: proofs of +Proposition 1.8 and Theorem 1.9 +In this section we give the proofs of Proposition 1.8 and Theorem 1.9. +Proof of Proposition 1.8. First we note that v := b − u > 0 is bounded and subharmonic outside +BR, in fact −∆v = ∆u = W ′(b − v) ≤ 0 in Rn\BR, hence by [12, Lemma 22] we have the decay +estimate +(3.39) +v(x) ≤ C|x|2−n +∀, |x| ≥ ρ. +Next, it follows from [13, Proposition 1, Theorem 4] that v is radial. +□ +Proof of Theorem 1.9. First we show that u < b in all Rn. By the strong maximum principle, it is +enough to prove that u ≤ b in Rn. For this purpose, assume by contradiction that c := supRn u = +maxRn u > b, and W(c) > W(b). Next, define the auxiliary potential +˜ +W(u) = + + + + + +W(b) +for u ≤ b +W(u) +for b ≤ u ≤ c +W(c) +for u ≥ c, +and consider a minimiser φR ∈ H1(BR(0)) of the energy functional +˜E(v) = +� +BR(0) +�1 +2|∇v(x)|2 + ˜W(v(x)) +� +dx, +in the class A = {v ∈ H1(BR(0)), v = c on ∂BR(0)}. We know that φR is a radial solution to (1.1) +such that b < minBR(0) φR = φR(0) < c, for R ≥ R0 sufficiently large, since φR(0) → b as R → ∞. +In addition, since u(Rn \ BR(0)) ⊂ (a, b), we have u(x + x0) < φR0(x) on BR0(0), provided that +|x0| > R + R0. Finally, by applying the sliding method of Berestycki, Caffarelli, and Nirenberg [1, +Lemma 3.1], we deduce that u ≤ φR0(0) < c holds on Rn, which is a contradiction. +So far we have established that W(c) = W(b). To conclude that u ≤ b, it remains to show that +c = b. Indeed, if c > b and W(c) = W(b), then c is a local minimum of W satisfying W ′(c) = 0, +and there exists x0 ∈ Rn such that u(x0) = c. Thus, by the maximum principle, we obtain u ≡ c, +which is excluded. +To complete the proof of Theorem 1.9, we shall use Proposition 1.8, [12, Theorem 2], and the +regularity of W. We first assume that hypothesis (i) holds, and distinguish the following cases. +a) If W ′′(b) > 0, then v := b − u > 0 is a decaying entire solution to +−∆v = f(v) := W ′(b − v), +therefore it is radial by [12, Theorem 2], since f ′(t) ≤ 0 for t ∈ (0, δ). +Otherwise, W ′′(b) = 0 implies that W ′′′(b) = 0, since W ∈ C6(R), thus we shall examine the +sign of d4W +du4 (b). +b) In the case where d4W +du4 (b) > 0, the radial symmetry of u follows again from [12, Theorem 2], +since f ′(t) ≤ 0 holds for t ∈ (0, δ). +c) In the case where d4W +du4 (b) = 0, we have d5W +du4 (b) = 0, and [b − δ, b] ∋ t �→ +W ′(t) +|t−b|5 is H¨older +continuous. Moreover, 5 ≥ n+2 +n−2 holds for every n ≥ 3, hence the result follows from Proposition +(1.8). +Finally, in the case where hypothesis (ii) holds, the result is straightforward in view of [12, +Theorem 2]. + +14 +MATTEO RIZZI AND PANAYOTIS SMYRNELIS +□ +4. A nonradial solution converging to the local maximum +In this section we will provide an example of a potential W of the form (1.7b) for which equation +(1.1) admits a solution u such that u(x) > a for |x| > R and lim|x|→∞ u(x) = a, but u is not radial. +The counterexample can be found in [6] using the Yamabe equation +(4.40) +− ∆u = n(n − 2) +4 +|u| +4 +n−2 u in Rn, n ≥ 3. +Equation (4.40) is variational, in the sense that it is the Euler-Lagrange equation of the energy +functional +E(u) := 1 +2 +� +Rn |∇u|2 − (n − 2)2 +8 +� +Rn |u| +2n +n−2 . +It is known that the only finite energy positive solutions are given by +µ− n−2 +2 U(µ−1(x − ξ)), +U(x) := +� +2 +1 + |x|2 +� n−2 +2 +, µ > 0, ξ ∈ Rn. +These solutions which are called the standard bubbles, are also the only positive solutions of (4.40) +(see [5]). +Using these bubbles, in [6] the authors construct a sequence of bounded entire solutions {uk}k≥k0 +to (4.40) in Rn of the form +(4.41) +uk := vk + φk, +where the approximate solution vk is given by +(4.42) +vk(x) := U(x) − +k +� +j=1 +µ +− n−2 +2 +k +U(µ−1 +k (x − ξj,k)), +µk = cnk−2 for n ≥ 4, µk = c3k−2(log k)−2 for n = 3 +ξj,k := (cos(2πj +k ), cos(2πj +k ), 0, . . . , 0) +1 ≤ j ≤ k +and the corrections φk fulfil +(4.43) +|φk(x)| ≤ +c +log k(1 + |x|) if n = 3, +|φk(x)| ≤ +c +kαn(1 + |x|n−2) if n ≥ 4, with αn > 0. +As a consequence, these solutions vk are L∞(Rn) close to a linear combination of k + 1 rescaled +bubbles. One of them is positive and centred at the origin, the other ones are negative and centred +along the unit circle S1 ⊂ R2. It particular, they are sign changing solutions. Moreover, it follows +from (4.42) and (4.43) that +(4.44) +uk(x) → 0 as |x| → ∞, for any k ≥ k0. +We are going to check that uk is positive outside a ball. +Lemma 4.1. There exist ¯r > 0 and ¯k > 0 such that uk(x) > 0 if |x| > ¯r and k ≥ ¯k. + +SOME RIGIDITY RESULTS +15 +Proof. We will show that, for k large enough, the approximate solution vk fulfils +(4.45) +vk(x) > 2 +3U(x) > 0 +if |x| > ¯r +for some large ¯r > 0. Then we apply (4.43) to conclude that +uk(x) = vk(x) + φk(x) > 2 +3U(x) − +C +(1 + |x|) log k > 1 +2U(x) > 0 +outside a large ball in dimension n ≥ 3. Similarly, in higher dimension we have +uk(x) = vk(x) + φk(x) > 2 +3U(x) − +C +kαn(1 + |x|n−2) > 1 +2U(x) > 0 +outside a large ball. +In order to prove (4.45), we note that, in dimension n = 3 we have +vk(x) = +� +2 +1 + r2 +� 1 +2 +− +k +� +j=1 +µ +− 1 +2 +k +U +�x − ξj,k +µk +� +≥ +� +2 +1 + r2 +� 1 +2 +− kµ +− 1 +2 +k +� +2µ2 +k +µ2 +k + (r − 1)2 +� 1 +2 +≥ +� +2 +1 + r2 +� 1 +2 � +1 − kµ +1 +2 +k +� 1 + r2 +(r − 1)2 +� 1 +2 � += +� +2 +1 + r2 +� 1 +2 � +1 − +√c3 +log k +� 1 + r2 +(r − 1)2 +� 1 +2 � +> 2 +3U(x) +where r = |x| and k are large enough. Similarly, in higher dimension, we have +vk(x) ≥ +� +2 +1 + r2 +� n−2 +2 +� +1 − c +n−2 +2 +n +kn−3 +� 1 + r2 +(r − 1)2 +� n−2 +2 � +> 2 +3U(x). +for r and k large enough. +□ +Finally, we can take b > ∥u¯k∥L∞(Rn) and define a C1(R) function f such that f(t) < 0 for any +t ∈ (0, b), f(t) = − n(n−2) +4 +|t| +4 +n−2 t for |t| ≤ ∥u¯k∥L∞(Rn), and f(b) = 0. Then f ′(0) = 0 and u¯k is a +solution to ∆u = f(u). Taking W to be a primitive of f, we have the required counter example. In +fact, we have 0 < u¯k(x) < b in D := Rn\B¯r, u¯k → 0 as |x| → ∞ but u¯k is sign changing and not +radial. +5. Appendix +5.1. The method of sub- and supersolutions. Let Ω ⊂ Rn be a open set with Lipschitz +boundary, and let f ∈ Cα +loc(R), for some α ∈ (0, 1). We say that u ∈ W 1,2(Ω) is a subsolution +(respectively u ∈ W 1,2(Ω) is a supersolution) to +(5.46) +∆u = f(u), +if ∆u ≥ f(u) (respectively ∆u ≤ f(u)) holds in Ω in the weak sense. +Proposition 5.1. Let u ≤ u be a couple of bounded W 1,2(Ω) sub- and supersolutions to (5.46). +Then, there exists a solution u ∈ C2(Ω) ∩ W 1,2(Ω) to (5.46), satisfying u ≤ u ≤ u. +Proof. We introduce the nonlinearity +(5.47) +g(x, u) := + + + + + +f(u(x)) +if u < u(x), +f(u) +if u(x) ≤ u ≤ u(x), +f(u(x)) +if u > u(x), + +16 +MATTEO RIZZI AND PANAYOTIS SMYRNELIS +and set G(x, u) = +� u +0 g(x, t)dt. Next, we establish (exactly as in the proof of [7, Lemma 1.1.1]), the +existence of a minimizer u of the energy functional: +(5.48) +E(v) = +� +Ω +�1 +2|∇v(x)|2 + G(x, v(x)) +� +dx, +in the class A = u + W 1,2 +0 +(Ω). +For the sake of simplicity, we consider in the definition of A, +the boundary condition v = u on ∂Ω. However, we could also set A = φ + W 1,2 +0 +(Ω), with any +φ ∈ W 1,2(Ω) such that u ≤ φ ≤ u holds on ∂Ω. By construction, u solves the Euler-Lagrange +equation +(5.49) +∆u = g(x, u), x ∈ Ω. +Moreover, it follows from the maximum principle that u ≤ u ≤ u in Ω, which yields that u is +actually a C2(Ω) solution to (5.46), satisfying u ≤ u ≤ u. +□ +Remark 5.2. If in Proposition 5.1, we consider a domain Ω = {x ∈ Rn : ρ1 < |x| < ρ2}, and a +couple u ≤ u of bounded radial sub- and supersolutions to (5.46), then we obtain the existence of a +radial solution u ∈ C2(Ω) ∩ W 1,2(Ω) to (5.46), satisfying u ≤ u ≤ u. Indeed, since the nonlinearity +(5.47) and the energy functional (5.48) are invariant by the orthogonal group O(n), we can look +for a minimizer u in the class AO(n) = {v ∈ A : v(σx) = v(x), ∀σ ∈ O(n)}. By the principle of +symmetric criticality [20], u is a smooth radial solution to (5.49), and the bounds u ≤ u ≤ u follow +as previously from the maximum principle. +The method of sub- and supersolutions is also applicable in unbounded domains. In Proposition +2.2, we apply it in Ω = Rn\Bρ, with a radial subsolution φ∗(x) = c|x|2−n, and a radial supersolution +φ∗ ≥ φ∗, φ∗ ∈ W 1,2(BR \ Bρ), ∀R > ρ. As a consequence of Proposition 5.1 and Remark 5.2, we +obtain for every R > ρ, a radial solution vR to (1.1) in ΩR := BR \ Bρ, satisfying +• φ∗ ≤ vR ≤ φ∗ in ΩR, +• vR = φ∗ on ∂ΩR. +In addition, since for any α ∈ (0, 1), the C1,α norm of ∂ΩR is uniformly bounded, and the C1,α +norm of φ∗ is also bounded in Ω, we deduce that the C1,α norm of vR is uniformly bounded in ΩR, +∀R > ρ (cf. [14, Theorem 8.33]). Finally, we use the Theorem of Ascoli, via a diagonal argument, +to prove that the limit v = limR→∞ vR exists (up to subsequence) and is a radial solution to (1.1) +in Ω, satisfying φ∗ ≤ v ≤ φ∗ in Ω. +In Proposition 2.6, we have a second application of the method of sub- and supersolutions in an +unbounded domain D, such that ∂D is bounded for the C1,α norm. Here again, we consider an +increasing sequence of bounded domains Dk, such that D = ∪kDk, and the boundaries ∂Dk are +uniformly bounded for the C1,α norm. In view of Proposition 5.1, we obtain in each domain Dk a +solution uk of (1.1), and then by taking the limit u = limk→∞ uk via the same diagonal argument, +we construct the solution u in the whole domain D. +5.2. Two lemmas for superharmonic functions. Here we recall two classical results on super- +harmonic functions. +Lemma 5.3. Let n ≥ 3, let Bρ ⊂ Rn be the open ball of radius ρ centered at the origin, and let +u ∈ C2(Rn \ Bρ) be a positive and bounded function, such that ∆u ≤ 0 in Rn\Bρ. Then, there +exists a constant c > 0 such that u(x) ≥ c|x|2−n, for any x ∈ Rn\Bρ. + +SOME RIGIDITY RESULTS +17 +Proof. We fix y ∈ Rn\Bρ(0), ε > 0 and we prove that u(y) ≥ c|y|2−n − ε, for some constant c > 0 +independent of ε, so that the result follows by letting ε → 0. +In order to do so, we note that +u(x) ≥ inf +∂Bρ u =: cρ2−n = c|x|2−n > c|x|2−n − ε +∀ x ∈ ∂Bρ. +Moreover, taking R > |y| large enough, we have +c|x|2−n − ε < 0 < u(x) +∀ x ∈ ∂BR. +As a consequence, using that c|x|2−n − ε is harmonic in the set A := {x ∈ Rn : ρ < |x| < R}, the +maximum principle yields that u ≥ c|x|2−n − ε in A. In particular we have u(y) ≥ c|y|2−n − ε. +□ +Lemma 5.4. Let Br(0) ⊂ R2 be the open ball of radius r centred at the origin, and let ψ ∈ +C(R2 \ Br(0)) be a function such that +• ψ ∈ W 1,2 +loc (R2 \ Br(0)), +• ψ is bounded from below on R2 \ Br(0), +• ∆ψ ≤ 0, on R2 \ Br(0). +Then, ψ attains its minimum on ∂Br(0). +Proof. Let x0 ∈ ∂Br(0) be such that min∂Br(0) ψ = ψ(x0). For every ǫ > 0 fixed, we consider the +function ζǫ(x) = ψ(x) + ǫ ln(|x|/r) which is superharmonic on R2 \ Br(0). In addition, we have +ζǫ(x) > ζǫ(x0) = ψ(x0), provided that |x| ≥ Rǫ (with Rǫ sufficiently large). Thus, by the maximum +principle, the minimum of ζǫ in the annuli r ≤ |x| ≤ R, with R ≥ Rǫ, is attained at x0. This implies, +that for every ǫ > 0, and x ∈ R2 \ Br(0), we have ζǫ(x) ≥ ψ(x0) ⇔ ψ(x) ≥ ψ(x0) − ǫ ln(|x|/r). +Finally, letting ǫ → 0, we obtain that ψ(x) ≥ ψ(x0) holds for every x ∈ R2 \ Br(0). +□ +Acknowledgements +M. Rizzi was partially supported by Justus Liebig University. +The authors are particularly +grateful to prof. Alberto Farina for his precious remarks and comments. +References +[1] H. Berestycki, L. Caffarelli, L. Nirenberg, Monotonicity for elliptic equations in an unbounded Lipschitz domain, +Comm. Pure Appl. Math. 50 (1997), 1089–1112. +[2] H. Berestycki, P.- L. Lions, Nonlinear scalar fields equations, I Existence of a ground state; II existence of +infinitely many solutions, Arch. Rational Mech. Analysis 82, 313-375 (1983) +[3] M. F. Bidaut-V´eron, S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems. +(English +[4] L. Caffarelli, N. Garofalo, F. Segala: A Gradient bound for entire solutions of quasi-linear equations and its +consequences. Commun. Pure Appl. Math. 47(11), 1457–1473 (1994). +[5] L. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations +with critical Sobolev growth. Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297. +[6] M. Del Pino, M. Musso, F. Pacard, A. Pistoia, Large energy entire solutions for the Yamabe equation. J. +Differential Equations 251 (2011), no. 9, 2568–2597. +[7] L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Monographs and Surveys in Pure and +Applied Mathematics 143, Chapman & Hall/CRC, Boca Raton (2011). +[8] L. Dupaigne, A. Farina, Liouville theorems for stable solutions of semilinear elliptic equations with convex +nonlinearities. Nonlinear Anal. 70 (2009), no. 8, 2882–2888. +[9] A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of RN, J. +Math. Pures Appl. (9) 87 (2007), no. 5, 537–561. + +18 +MATTEO RIZZI AND PANAYOTIS SMYRNELIS +[10] A. Farina, Rigidity and one-dimensional symmetry for semilinear elliptic equations in the whole of RN and in +half spaces. Adv. Math. Sci. Appl. 13 (2003), no. 1, 65–82. +[11] A.Farina, Symmetry for solutions of semilinear elliptic equations in RN and related conjectures. Papers in +memory of E. De Giorgi, Ricerche Mat., 48 (1999), suppl., 129–154. +[12] A. Farina, A. Malchiodi, M. Rizzi: Symmetry properties of some solutions to some semilinear elliptic equations, +Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), no. 4, 1209–1234. +[13] B. Gidas, W. M. Ni, L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in Rn. +Mathematical analysis and applications, Part A, pp. 369–402, Adv. in Math. Suppl. Stud., 7a, Academic Press, +New York-London, 1981. +[14] Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations of second order. Grundlehren der mathe- +matischen Wissenschaften 224, Springer-Verlag, Berlin, revised second edition, (1998). +[15] A. Hernandez, M. Kowalczyk, Rotationally symmetric solutions to the Cahn-Hilliard equation Discrete Contin. +Dyn. Syst. 37 (2017), no. 2, 801–827. +[16] J. Hirata, N. Ikoma, K. Tanaka, Nonlinear scalar field equations in Rn: mountain pass and symmetric mountain +pass aproaches, Topological Methods in Nonlinear Analysis, Journal of the Juliusz Schauder Center 35 (2010) +253–276. +[17] M. Kowalczyk, M. Rizzi, Multiple Delaunay ends solutions of the Cahn-Hilliard equation, Comm. Partial +Differential Equations 47 (2022), no. 4, 829–874. +[18] C. T. Ledesma, C´esar T, Y. Ma, Y. Wang, Lane-Emden equations perturbed by nonhomogeneous potential in +the super critical case. (English summary) Adv. Nonlinear Anal. 11 (2022), no. 1, 128–140. +[19] Modica, L.: A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. +Math. 38, 679–684 (1985). +[20] Palais, R. S.: The principle of symmetric criticality. Commun. Math. Phys. 69 no. 1, 19–30 (1979) +[21] M. Rizzi, Radial and cylindrical symmetry of solutions to the Cahn-Hilliard equation, Calc. Var. Partial +Differential Equations 59 (2020), no. 2, Paper No. 75, 13 pp. +(M. Rizzi) Mathematisches Institut, Justus Liebig Universit¨at, Arndtstrasse 2, 35392, Giessen, Ger- +many. +Email address: mrizzi1988@gmail.com +(P. Smyrnelis) Department of Mathematics, University of Athens, 11584 Athens, Greece +Email address, P. Smyrnelis: smpanos@math.uoa.gr + diff --git a/QNFRT4oBgHgl3EQf6zgQ/content/tmp_files/load_file.txt b/QNFRT4oBgHgl3EQf6zgQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..60a7ef2d646609a9730ded58837f703913b52b2d --- /dev/null +++ b/QNFRT4oBgHgl3EQf6zgQ/content/tmp_files/load_file.txt @@ -0,0 +1,893 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf,len=892 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='13677v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='AP] 31 Jan 2023 SOME RIGIDITY RESULTS AND ASYMPTOTIC PROPERTIES FOR SOLUTIONS TO SEMILINEAR ELLIPTIC P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' MATTEO RIZZI AND PANAYOTIS SMYRNELIS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' We will present some rigidity results for solutions to semilinear elliptic equations of the form ∆u = W ′(u), where W is a quite general potential with a local minimum and a local maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' We are particularly interested in Liouville-type theorems and symmetry results, which generalise some known facts about the Cahn-Hilliard equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' MSC2020: 35B06;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' 35B50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' 35B53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Keywords: Liouville theorems, radial symmetry, rigidity, Cahn-Hilliard equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Introduction and main results Our aim is to revisit some asymptotic properties and rigidity results for solutions u to the semilinear elliptic P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) ∆u(x) = W ′(u(x)), u ∈ C2(D), D ⊂ Rn, n ≥ 2, with a potential W ∈ C1,1 loc (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' The domains D ⊂ Rn we consider are connected open unbounded sets such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='2) ∀R > 0, D contains a ball of radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' We shall also assume that on D the solution u takes its values in a bounded interval where W is monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' For instance, one may consider the Cahn-Hilliard equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='3) ∆u = W ′(u) = u3 − u + δ in Rn, with |δ| < 2 3 √ 3, so that the polynomial f(t) = t3 − t + δ admits exactly three real roots z1(δ) < − 1 √ 3 < z2(δ) < 1 √ 3 < z3(δ), and is negative on the interval (z2(δ), z3(δ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' This equation was largely studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' For example, some particular solutions were constructed in [15] and [17], while some results about radial and cylindrical symmetry of solutions and Liouville type results can be found in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Our starting point is the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' [[21]] Let n ≥ 2, δ ∈ (− 2 3 √ 3, 2 3 √ 3) and let uδ be a solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='3) such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='4) uδ > z2(δ) outside a ball BR ⊂ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' (1) If δ ∈ (− 2 3 √ 3, 0], then u ≡ z3(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' (2) If δ ∈ (0, 2 3 √ 3), then uδ is radially symmetric (not necessarily constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' 1 2 MATTEO RIZZI AND PANAYOTIS SMYRNELIS Similar results in the case δ = 0 can be found in [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' The purpose here is to extend Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1 to more general non linearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' The proofs in [21] are based on some known symmetry results (see [12]) which rely on the mov- ing planes method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' A key tool in these methods is the maximum principle, even for unbounded domains (see [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' If (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='5) u(D) ⊂ [a, b], and W ′ < 0, on [a, b) (with a, b ∈ R), it is straightforward by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1 below that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='6) lim d(x,∂D)→∞ u(x) = b, and W ′(b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Thus, we shall focus on the more involved problem where (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='7a) u(D) ⊂ (a, b], (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='7b) W ′(a) = W ′(b) = 0, and W ′ < 0, on (a, b) (with a, b ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' If we assume in addition to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='7), the nondegeneracy condition: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='8) W ′(s) ≤ −C0(s − a) on [a, s0], for some C0 > 0 and s0 ∈ (a, b), we can apply comparison arguments of Berestycki, Caffarelli, and Nirenberg [1, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='2] to deduce that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='9) d(x, ∂D) > η ⇒ u(x) ≥ a + ǫ, for some constants η, ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Consequently, the asymptotic property (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='6) follows again from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' On the other hand, in the degenerate case where (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='8) does not hold, the asymptotic behaviour of the solutions may be more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In the case where D is the complement of a ball, we can relax condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='8) by assuming that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='10) W ′(s) ≤ −C0(s − a)p on [a, s0], for some C0 > 0, p < n n − 2, and s0 ∈ (a, b), Under assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='10), we can still prove the asymptotic property (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='6) for solutions provided (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='7) holds (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' However, in the case of potentials such that lim u→a+ W ′(u) (u − a)p = −λ for some λ > 0 and p > n n−2, radial solutions u : Rn → (a, b) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='11) lim |x|→∞ u0(x) = a may exist in dimensions n ≥ 3 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Therefore, condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='10) is optimal to derive (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='6), when D is the complement of a ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' For general domains, condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='10) is not sufficient to deduce the asymptotic behaviour of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='6, we construct a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) in a dumbbell shaped domain D ⊂ R2, such that u ≈ a on the one side of the neck, while u ≈ b on the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' To sum up these results, we now state SOME RIGIDITY RESULTS 3 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Let W ∈ C1,1 loc (R) be a potential satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='7b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' (i) Assume u ∈ C2(Rn) is a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) such that u(Rn) ⊂ [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Then, when n = 2, or n ≥ 3 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='10) holds, we have either u ≡ a, or u ≡ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Otherwise (when n ≥ 3 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='10) does not hold), we have either u ≡ a, or u ≡ b, or1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='12) � u(Rn) ⊂ (a, CW ], for a constant CW ∈ (a, b) depending only on W, lim inf|x|→∞ u(x) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' (ii) Assume the domain D satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='2), and u ∈ C2(D) is a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) such that u(D) ⊂ (a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Then, we have limd(x,∂D)→∞ u(x) = b, provided that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='8) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Next, we derive some Liouville type results by considering domains D ⊂ Rn satisfying the following condition: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='13) the radii of the balls contained in Rn \\ D are uniformly bounded by a constant Λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Let W ∈ C1,1 loc(R) be a non negative potential satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='7b), and W(b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Assume the domain D satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='13), and u ∈ C2(Rn) is a bounded entire solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) such that supRn u = b, and u(D) ⊂ (a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Then, u ≡ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Modica [19] proved that if W ∈ C2(R) is a non negative potential, and u is a bounded solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) in Rn, then the condition W(u(x0)) = 0 for some x0 ∈ Rn implies that u is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In the sequel, a new proof of this result which also applies to potentials W ∈ C1,1 loc(R) was proposed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Therefore, the hypothesis supRn u = b in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='3 is not very strong, since the condition u(D) ⊂ (a, b] yields that either u < b in Rn or u ≡ b, so that supRn u ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Since the linear behaviour of W ′ near the local maximum (see condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='8)) implies that limd(x,∂D)→∞ u(x) = b, when D satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='2) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='2 (ii)), we obtain a first corollary of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='3: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Let W ∈ C1,1 loc(R) be a non negative potential satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='7b), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='8) and W(b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Assume the domain D satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='2) as well as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='13), and u ∈ C2(Rn) is a bounded entire solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) such that u(D) ⊂ (a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Then, u ≡ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Finally, we particularise Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='5 in the case where D is the complement of a ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Let W ∈ C1,1 loc (R) be a nonnegative potential satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='7b), and W(b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Assume u ∈ C2(Rn) is an entire solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='14) u(x) ∈ (a, b] ∀ x ∈ Rn\\BR, for some R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Then, u ≡ b, provided that n = 2, or n ≥ 3 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='10) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' We will prove these results in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='6 was established in [21] for entire solutions to the Cahn-Hilliard equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Here we extend this result to general nonlinearities under optimal assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Indeed, the necessity of condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='10) (when n ≥ 3) for Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='6 to hold, is clear in view of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' 1For instance, let u0 be the radial solution provided by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Then, by taking u(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' , xn, xn+1) = u0(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' , xn), we can see that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='12) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' 4 MATTEO RIZZI AND PANAYOTIS SMYRNELIS Other Liouville type results for stable solutions to semilinear PDEs were established in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Here there is no stability assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' After that, we will address the issue of radial symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In [13], the authors prove radial symmetry of solutions to fully nonlinear equations of very general form, provided these solutions have a suitable asymptotic polynomial decay at infinity (see Theorem 4 there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Here we are in- terested in radial symmetry of solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) with W satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='7b), assuming that either lim|x|→∞ u(x) = b or lim|x|→∞ u(x) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' The case in which lim|x|→∞ u(x) = b is easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' The following result is a consequence of [13, Proposition 1]: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Let u : Rn → R be a solution to ∆u = W ′(u), n ≥ 3, where W ∈ C2(Rn) is a potential fulfilling (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='7b) and such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='15) [b − δ, b] ∋ t �→ W ′(t) |t − b|p is H¨older continuous for some δ > 0, p ≥ n + 2 n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Assume that u < b in Rn and lim|x|→∞ u(x) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Then u is radially symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' On the other hand, if W is convex in an interval (b−δ, b), then the symmetry result follows from [12, Theorem 2] in any dimension n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='15) is not required anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In view of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='8 and [12, Theorem 2], we obtain the following generalisation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1 (ii): Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Let W ∈ C2(R) be a potential such that W ′(t) < 0 for any t ∈ (a, b), W ′(a) = 0, and W(t) ≥ W(b) for any t > b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In addition, we suppose that one of the following is true: (i) n ≥ 3, and W ∈ C6,α(b − δ, b + δ), for some δ > 0 and α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' (ii) n ≥ 2, and W is convex in (b − δ, b), for some δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Assume also that u : Rn → R is a solution to ∆u = W ′(u) such that u(Rn\\BR) ⊂ (a, b) and lim|x|→∞ u(x) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Then u < b in Rn and it is radially symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' (1) If a potential W satisfies the assumptions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='9, then it has a local minimum at t = b, so that W ′(b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' However, this minimum is not required to be a global one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' (2) If W ′(t) > 0 for t > b, it follows from the maximum principle that any bounded solutions u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) in Rn, satisfies the bound u ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' (3) Let u be a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Then, if n = 2 or n ≥ 3 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='10) holds, the condition u(Rn \\ BR) ⊂ (a, b), implies that lim|x|→∞ u(x) = b in view of Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' (4) For the existence of radial solutions satisfying lim|x|→∞ u(x) = b, we refer to [2, Theorem 1, Theorem 4] and [16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' We will prove these symmetry results in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Assuming again that W satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='7b), the description of entire solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) converging to a at infinity is a much more difficult task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In that case, only a few symmetry results are available, under somewhat restrictive hypotheses on the solution and the nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Some results can be found in [5], where a monotonicity assumption is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' As a particular case, their results apply to bounded solutions to the Lane-Emden equation −∆u = |u|p−1u SOME RIGIDITY RESULTS 5 in Rn, for which several Liuoville type results are known (see for example [3, 9, 18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' For future purposes, the main difficulty is to remove the monotonicity and convexity assumption about the non-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='2, we know that non trivial solutions can exist only if condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='10) is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' However, the fact that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='16) u(Rn\\BR) ⊂ (a, b) cannot guarantee a Liouville type result (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='3), or even radial symmetry under the assumption that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='17) lim |x|→∞ u(x) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In section 4, we check that the solutions constructed in [6], provide examples of nonradial solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1), such that u(x) − a changes sign in a compact set, and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='16) as well as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='17) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' It would be interesting to see if a nonradial solution satisfying u(Rn) ⊂ (a, b) and lim|x|→∞ u(x) = a, may also exist for a potential W having a negative derivative on the range of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' To the best of our knowledge, this is a difficult open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Asymptotic behaviour and Liouville type results We first prove a basic lemma on the asymptotic behaviour of solutions satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Let D ⊂ Rn be a domain satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='2), and let u be a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) (W ∈ C1,α loc (R), α ∈ (0, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Assume also that u(D) ⊂ [a, b], and W ′ < 0 on the interval [a, b) (with a, b ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Then, limd(x,∂D)→∞ u(x) = b, and W ′(b) = 0 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' If in addition D = Rn, then we have u ≡ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' We first recall that for fixed R > 0, the solution u is uniformly bounded in C2,α (for some α ∈ (0, 1)) on the balls BR(x) satisfying d(x, ∂D) > R + 1 with x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Let l := lim infd(x,∂D)→∞ u(x), and let {xk} ⊂ D be a sequence such that limk→∞ d(xk, ∂D) = ∞, and limk→∞ u(xk) = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' We set vk(y) = u(xk + y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In view of the previous estimates, we can apply the Ascoli theorem via a diagonal argument to the sequence {vk}, and deduce that up to subsequence, vk converges in C2 loc(Rn) to an entire solution v∞ of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Moreover, we have v∞(0) = l = min y∈Rn v∞(y), and 0 ≤ ∆v∞(0) = W ′(l) ≤ 0, so that, l = b, W ′(b) = 0, and v∞ ≡ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' This proves that limd(x,∂D)→∞ u(x) = b, and W ′(b) = 0 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In the particular case where D = Rn, we have u ≡ b, since otherwise u would attain its minimum at a point x0 where 0 ≤ ∆u(x0) = W ′(u(x0)) < 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' □ Next, given a potential satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='7b), we study the existence of solutions such that u(Rn) ⊂ (a, b), for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' The answer to this question depends on the growth of W ′ in a right neighbourhood of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='2 below, we first examine the case of potentials for which (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='10) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Let n ≥ 3, let Bρ ⊂ Rn be the open ball of radius ρ centred at the origin, and let W ∈ C1,1 loc(R) be a potential fulfilling (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='7b), and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Then, every solution u ∈ C2(Rn \\ Bρ) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) such that u(Rn \\ Bρ) ⊂ (a, b), satisfies lim|x|→∞ u(x) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' 6 MATTEO RIZZI AND PANAYOTIS SMYRNELIS Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Without loss of generality, we may assume that a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Assume by contradiction that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='18) u(Rn \\ Bρ) ⊂ (0, b − η], for some η > 0 small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Then, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='19) W ′(u) ≤ −c1up, ∀u ∈ [0, b − η] for a constant 0 < c1 < C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' We first examine the case where u is radial, that is, u(x) = v(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' As a consequence, v solves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='20) v′′(r) + n − 1 r v′(r) = W ′(v(r)), ∀r ∈ [ρ, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Our claim is that v′(ρ0) ≤ 0 holds for some ρ0 ≥ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Indeed, otherwise, we would have ∀r ∈ [ρ, ∞) : v′(r) > 0, and v′′(r) ≤ κ := max [v(ρ),b−η] W ′ < 0, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' So far, we have proved that v′(ρ0) ≤ 0 for some ρ0 ≥ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' By noticing that v′(ρ0) = 0 ⇒ v′′(ρ0) < 0 in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='20), one can see that v′ < 0 holds on an interval (ρ0, ρ0 + ǫ), for small ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Let l := sup{r > ρ0 : v′ < 0 on (ρ0, r)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' It is clear that l = ∞, since otherwise we would deduce that v′(l) = 0 and v′′(l) < 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' This establishes that v′ < 0 on (ρ0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Now, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='20) that ∀r > ρ1 : rn−1v′(r) ≤ rn−1v′(r) − ρn−1 0 v′(ρ0) = � r ρ0 sn−1W ′(v(s))ds ≤ −c1vp(r) � r ρ0 sn−1ds ≤ −kvp(r)rn, for a constant k > 0, and for ρ1 > ρ0 large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Next, an integration of the previous inequality gives ∀r > ρ1 : v1−p(r) ≥ v1−p(r) − v1−p(ρ1) ≥ k(p − 1) 2 (r2 − ρ2 1), from which we deduce that v(r) ≤ ˜kr− 2 p−1 , for a constant ˜k > 0, and for r > ρ2 > ρ1 large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Since p < n n−2 ⇔ 2 p−1 > n − 2, this contradicts the lower bound provided by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Therefore the existence of a radial solution satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='18) is ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' To complete the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='2, we also have to exclude the existence of non radial solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Assume by contradiction that u ∈ C2(Rn \\ Bρ) is a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In view of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='3, u satisfies the lower bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='21) u(x) > φ∗(x) = c|x|2−n, c > 0, where φ∗ is a subsolution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1), that is, ∆φ∗ = 0 ≥ W ′(φ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Starting from u, we shall construct a radial supersolution φ∗ of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1), such that φ∗ ≤ φ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Let ρi,m be the rotation of angle π 2m around the xi coordinate axis of Rn (m ≥ 1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' , n − 1), and let Gm := {ρk1 1,m ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' ◦ ρkn−1 n−1,m : 0 ≤ ki ≤ 2m+1 − 1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Using spherical coordinates, one can see that given |x0| ≥ ρ, the set ∪m≥1Gmx0 is dense in the sphere {x ∈ Rn : |x| = |x0|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In particular, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='22) lim m→∞ min g∈Gm u(gx0) = min |x|=|x0| u(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' SOME RIGIDITY RESULTS 7 Next, we notice that for every g ∈ Gm, x �→ u(gx) solves (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' On the other hand, in view of the Kato inequality, φm(x) := ming∈Gm u(gx) is a supersolution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1), satisfying φ∗ ≤ φm ≤ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In addition, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='22) that φ∗(x) := limm→∞ φm(x) = min{u(y) : |y| = |x|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Finally, since |∇φm| is uniformly bounded on Rn\\Bρ, we obtain that (up to subsequence) φm conververges weakly to φ∗ in W 1,2(BR \\ Bρ), for every R > ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' This implies that φ∗ (which belongs to W 1,2(BR \\ Bρ), for every R > ρ) is a radial supersolution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) satisfying φ∗ ≤ φ∗ ≤ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' To conclude, we deduce from the method of sub- and supersolutions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1, and for instance [7, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1]), the existence of a radial solution v ∈ C2(Rn \\ Bρ), satisfying 0 < φ∗ ≤ v ≤ φ∗ ≤ b − η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In view of the first part of the proof, this is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' So far, we have established that every solution u ∈ C2(Rn \\ Bρ) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) such that u(Rn \\ Bρ) ⊂ (0, b), satisfies supRn\\Bρ u = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' That is, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='23) ∃{xk}k∈N : lim k→∞ |xk| = ∞, and lim k→∞ u(xk) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Setting vk(y) := u(xk + y), and proceeding as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1, we obtain that (up to subsequence) vk converges in C2 loc(Rn) to an entire solution v∞ of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Furthermore, since v∞(0) = b, the maximum principle implies that v∞ ≡ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' At this stage we consider a minimizer φR ∈ H1(BR(0)) of the energy functional (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='24a) ˜E(v) = � BR(0) �1 2|∇v(x)|2 + ˜W(v(x)) � dx, in H1 0(BR(0)), where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='24b) ˜W(v) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 W(a) for v ≤ 0 W(v) for 0 ≤ v ≤ b W(b) for v ≥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' It is known that φR is a smooth radial solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) in BR(0), such that 0 ≤ φR ≤ maxBR(0) φR := b − δR on BR(0), for some δR > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In addition, we have limR→∞ δR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Thus, given ǫ > 0, we can ensure that δR < ǫ for some R > 0 large enough, and φR ≤ b − δR ≤ vk holds on BR(0), for k ≥ kR large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Finally, by applying the sliding method of Berestycki, Caffarelli, and Nirenberg [1, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1], we deduce that u(x) ≥ φR(0) ≥ b − ǫ, provided that |x| > ρ + R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' This completes the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='2 □ In the subcritical case where W ′(u) ∼ −λ|u − a|p near a, with λ > 0 and p ∈ ( n n−2, n+2 n−2), we shall see in Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='4 below, that depending on the potential, there may or may not exist a radial solution such that u(Rn) ⊂ (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Given any n ≥ 3, p > n n−2 and λ > 0, there exists a potential W ∈ C2(R) fulfilling (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='7b), and a solution u ∈ C∞(Rn) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1), such that a) limu→a+ W ′(u) |u−a|p = −λ, b) u is radial and radially decreasing (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' u(x) = ˜u(|x|), for a smooth decreasing function ˜u : [0, ∞) → (a, b)), c) u(Rn) ⊂ (a, b), and lim|x|→∞ u(x) = a, d) W ′′(u(0)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' 8 MATTEO RIZZI AND PANAYOTIS SMYRNELIS Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Without loss of generality, we may assume that a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' First, we note that the function v(x) = � 2((n−2)p−n) λ(p−1)2 � 1 p−1 |x|− 2 p−1 solves the equation ∆v = −λvp in Rn\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Next, in order to eliminate the singularity at the origin, we take a smooth cutoff function ξ : R → [0, 1] such that \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ξ = 1 in [3, ∞), 0 < ξ < 1 and ξ′ > 0 in (2, 3), ξ = 0 in (−∞, 2], and we consider a function ˜u : (1, ∞) → R such that � ˜u′′(r) = ξ(r)˜v′′(r) ∀ r ∈ [1, ∞), ˜u(r) = ˜v(r) ∀r ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' where v(x) =: ˜v(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' One can see that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='25) ˜u′′ + n − 1 r ˜u′ < 0 in [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' The latter inequality is clear if r ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In order to prove that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='25) holds in [1, 3) too, we note that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='26) ˜u′(r) = − � ∞ r ˜u′′(t)dt = − � ∞ r ξ(t)˜v′′(t)dt = ξ(r)˜v′(r) + � ∞ r ξ′(t)˜v′(t)dt < ξ(r)˜v′(r) ≤ 0, ∀ r ∈ [1, 3), so that ˜u′′ + n − 1 r ˜u′ < ξ � ˜v′′ + n − 1 r ˜v′� ≤ 0, ∀ r ∈ [1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Now, we extend ˜u to a smooth even positive function on the whole R, still denoted by ˜u, fulfilling ˜u′ < 0 in (0, ∞), ˜u′′ < 0 in [0, 1), so that ˜u′′ + n−1 r ˜u′ < 0 holds in [0, ∞), ˜u′′′(0) = 0 and ˜u(4)(0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' This can easily be done if we recall that ˜u is affine and decreasing on [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Since ˜u is monotone in [0, ∞), then it is invertible in this interval with inverse function β : (0, ˜u(0)] → [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Finally, setting ϕ(r) := ˜u′′(r) + n − 1 r ˜u′(r), ∀ r > 0, and H(s) := ϕ(β(s)), for s ∈ (0, ˜u(0)], one can see that u(x) := ˜u(|x|) satisfies the equation ∆u = H(u) in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' We also notice that H(˜u(0)) = n˜u′′(0) < 0 and H′(˜u(0)) = (n+2)˜u(4)(0) 3˜u′′(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Thus, one can find a C1 extension of H to the whole R, still denoted by H, such that H < 0 in (0, b), for some b > ˜u(0), and H(b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' By construction, we have H(u) = −λup in (0, ˜u(3)), so that H(0) = H′(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In order to conclude the proof it is enough to define W to be the primitive of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Given any n ≥ 3, p ∈ ( n n−2, n+2 n−2), and λ > 0, there exists a potential W ∈ C2(R) fulfilling (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='7b) and limu→a+ W ′(u) |u−a|p = −λ, for which there are no radial solutions u ∈ C2(Rn) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) such that u(Rn) ⊂ (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' SOME RIGIDITY RESULTS 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Without loss of generality, we may assume that a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' We consider the function H(u) = −λup on an interval [0, β], and since p ∈ ( n n−2, n+2 n−2), we set ǫ = n p+1 − n−2 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' One can find a C1 extension of H to the whole R, still denoted by H, such that H < 0 in (0, b), and H(b) = 0, for some b > β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Let b = κβ, with κ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' H([0, b]) = [−λµβp, 0] for some µ > 1, such that κµ < 1 + 2ǫ n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Next, define W ∈ C2(R) to be the primitive of H vanishing at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' We claim that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='27) n − 2 2 W ′(u)u − nW(u) > 0 on (0, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Indeed, we have n−2 2 W ′(u)u − nW(u) = ǫλup+1 on [0, β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' On the other hand, if u ∈ [β, b], then it follows that n−2 2 W ′(u)u − nW(u) ≥ n−2 2 W ′(u)u − nW(β) ≥ ( n p+1 − n−2 2 κµ)λβp+1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Now that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='27) is established, we consider a radial solution u ∈ C2(Rn) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) such that u(Rn) ⊂ (0, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Setting v(|x|) = u(x) and proceeding as in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='2, one can see that v satisfies the standard estimates v(r) = O(r− 2 p−1 ), v′(r) = O(r− p+1 p−1 ), and W(v(r)) = O(r− 2(p+1) p−1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' To conclude we use the well-known Pohozaev identity: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='28) � r 0 sn−1(n − 2 2 W ′(v(s))v(s)−nW(v(s)) � ds = n − 2 2 rn−1v(r)v′(r)+rn�|v′(r)|2 2 −W(v(r)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' We notice that since p ∈ ( n n−2, n+2 n−2), the right hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='28) goes to 0, as r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' On the other hand, the left hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='28) is strictly positive in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' This rules out the existence of radial solutions such that u(Rn) ⊂ (0, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' □ The next Proposition examines the existence of radial solutions in the different regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Let n ≥ 3, and let W ∈ C1,1 loc(R) be a potential satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='7b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' (i) If (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='10) holds, there are no radial solutions u ∈ C2(Rn) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) such that u(Rn) ⊂ (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' (ii) If lim supu→a+ |W ′(u)| |u−a| n+2 n−2 = 0 holds, there exists a radial solution u ∈ C2(Rn) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) such that u(Rn) ⊂ (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' (iii) Otherwise, if neither (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='10) nor lim supu→a+ |W ′(u)| |u−a| n+2 n−2 = 0 hold, depending on W, there may or may not exist a radial solution u ∈ C2(Rn) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) such that u(Rn) ⊂ (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' (i) A radial solution u ∈ C2(Rn) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) such that u(Rn) ⊂ (a, b), decays to a, as |x| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In view of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='2, it is clear that such a solution does not exist when (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='10) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' (ii) Now, assume that lim supu→a+ |W ′(u)| |u−a| n+2 n−2 = 0 holds, and define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='29) ˜ W(v) = � W(v) for v ≤ b W(b) for v ≥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Theorem 4 of [2] provides the existence of a radial solution u ∈ C2(Rn) of ∆u = ˜W ′(u), such that u > a, and lim|x|→∞ u(x) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' By the maximum principle, we have u(Rn) ⊂ (a, b), and thus u solves ∆u = W ′(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Finally, (iii) follows from Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' □ 10 MATTEO RIZZI AND PANAYOTIS SMYRNELIS As we mentioned in the Introduction, for general domains, condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='10) is not sufficient to derive the asymptotic property (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='6) of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='6 below, provides examples of solutions having a different asymptotic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Let p > 1, and let W ∈ C1,1 loc(R) be a potential fulfilling (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='7b), as well as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='30) ∀u ∈ [a, b] : W ′(u) ≥ −c(u − a)p, for a constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Let D = {x ∈ R2 : |x2| < ψ(x1)}, where ψ ∈ C∞(R) is a positive function such that ψ(s) = λ|s|, for |s| > ǫ (with λ , ǫ > 0 sufficiently small, depending on W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Then, there exists a solution u ∈ C2(D) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) such that u(D) ⊂ (a, b), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='31) lim x1→+∞ u(x) = a and lim x1→−∞ u(x) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Without loss of generality we may assume that a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' We shall first construct a supersolution φ∗ of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' We define the auxilliary functions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='32a) f(reiθ) = r− 2 p−1 g(θ), with g : [−θ0, θ0] → (0, ∞) (θ0 < π 2 ), a positive solution of the O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' : (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='32b) g′′(θ) = −cgp(θ) − 4 (p − 1)2 g(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Next, setting λ = tan(θ0), one can check that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='33) ∆f(x) = −c(f(x))p in the sector S = {x1 > 0, |x2| < λx1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' In addition, we have f(x) > b in the set {0 < x1 ≤ ǫ, |x2| < λx1}, provided that ǫ > 0 is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Finally, we take (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='34) φ∗(x) = � min(f(x), b) when x1 > ǫ, and |x2| < λx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' b when x1 ≤ ǫ, and |x2| < ψ(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Using the Kato inequality, one can see that φ∗ is a supersolution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='1) in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content=' Indeed, in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='30) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='33), we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf'} +page_content='35) ∆φ∗ ≤ −cf pχ{f 100. For simplicity, +all chains have length L = 1.5 mm, and representative +disorder is introduced via their motion. +Their speeds +are constant in time, but initially drawn from a normal +distribution with average v0 = 3 µm s−1 and standard de- +viation 0.7 µm s−1, in accordance with experimental ob- +servations (Fig. 2(a)). +The position ri,α of bead α of +chain i follows the track laid out by its head, so that +ri,α(t) = ri,α−1(t − ∆t). At each time step ∆t the end +bead is removed from the tail of each chain, and a new +bead is added at its head, displaced by a distance vi∆t +at an angle θi (see Fig. S4). Similar models have been +applied to isolated filaments [22] and filaments moving on +a lattice [45]. Although this system has some similarities +to active polymers [13, 15–17, 46], those lack a unique +curvature autocorrelation time and instead each portion +of the polymer fluctuates independently; in contrast, the +fluctuations and curvature of our chains are solely deter- +mined by their heads and not by each segment thereof. +Motivated by the models of active nematic particles +used to simulate microtubules [26, 47], C. elegans [48] +and Pseudanabaena sp. [49], we now introduce a model of +interacting active chains, as appropriate to the behavior +of filamentous cyanobacteria. +Here, the orientation θi +and angular velocity ωi of the head of each chain i evolves +according to a modified Ornstein–Uhlenbeck process +dωi +dt = −1 +τ +dθi +dt + +� +2Dωξi(t), +(1) +dθi +dt = ωi − J +Nij +� +j∼i +∂ +∂θi +U(θi, θj), +(2) +where τ is the curvature autocorrelation time, Dω is the +orientational diffusion coefficient, and ξi(t) is a Gaussian +white noise with zero mean and unit variance. +Dω is +not directly accessible experimentally, but can be linked +to other physical parameters. +Without any filament- +filament interactions, Eq. (1) produces a normal distri- +bution of angular velocities with zero mean and variance + +3 +(d) +(a) +(b) +(c) +FIG. 2. Filament behavior. (a) The distribution of experi- +mentally observed gliding speeds (blue) is well-fit by a Gaus- +sian (red, used for simulations). +(b) Interacting filaments +align with a probability that depends on their angle of inci- +dence (histogram, left axis; interaction frequency, right axis). +In the model, an incident chain is deflected on average by the +relative angle ∆θ/θ (black line). (c) In bundles, the direc- +tions of filament motion have a nematic distribution: a polar +histogram compares experimental (blue) and simulated (red) +cases. (d) Schematic of a modeled pairwise interaction. When +the head of a chain is within distance d of another chain, it +experiences an aligning effect. +⟨ω2⟩ = Dωτ. For chains moving at speed v0 this trans- +lates into a curvature distribution with standard devi- +ation δκ = +� +⟨ω2⟩/v0. +Hence, Dω = (v0δκ)2/τ. +Fi- +nally, interactions of strength J are modeled by a ne- +matic Lebwohl–Lasher potential, U = − cos [2(θi − θj)], +averaged over the Nij chains within an interaction range +d of the head of chain i, where θj is the orientation of the +nearest bead on chain j (see Fig. 2(d)). +The model parameters were matched to experimen- +tal values of relatively isolated cyanobacteria, and fine- +tuned based on the collective behavior at higher densities. +In particular, unless otherwise stated, we set τ = 480 s, +δκ = 200 m−1 (resulting in Dω = 7.5 × 10−10 s−3) and +d = 5 µm, close to the observed values of 470 s, 340 m−1 +and the filament diameter of 4.2 µm, respectively. The +interaction strength, J = 0.006 s−1, was chosen by con- +sidering two filaments meeting at an angle θ. On average, +the effects of interactions are comparable if the incident +filament is either deflected by a relative angle ∆θ/θ, or by +the whole angle θ with probability palign(θ). As shown in +Fig. 2(b), in this sense J gives a similar average response +to the experimentally observed interactions. +Experimentally, colonies of cyanobacteria filaments are +(g) +(a) +10 mm +(c) +(f) +(h) +(d) +(b) +0 +20 +40 +60 +80 +Filament density ρ(mm−2) +0 +1 +� +S +� +(i) +sim. +exp. +(e) +FIG. 3. +Collective behavior and disorder-order transition. +Panels (a–d) show micrographs of colonies at densities ρ = +25, 31, 42, and 59 mm−2, respectively. +Panels (e–h) show +snapshots of simulations at comparable densities of ρ = +24, 31, 41 and 59 mm−2, respectively. To avoid boundary ef- +fects, simulations were performed in systems with sides 4.5× +larger than shown; the panels are cropped to match the mi- +crograph size. (i) Order parameter ⟨S⟩, averaged over 1 mm2 +blocks covering the experimental (blue) or simulated (red) +domain. Error bars and shading give the standard deviation +of ⟨S⟩ over the blocks. At low ρ the filaments are randomly +aligned, but locally-nematic bundles and a reticulated struc- +ture emerge above ρ ∼ 40 mm−2. +disordered at low density, but show emergent patterns at +higher densities, as shown in Fig. 3(a–d). The simulated +chains order in a similar way, Fig. 3(e–h), with reticulated +structures appearing at higher ρ. We quantify local order +in the steady-state by means of the 2D nematic order pa- +rameter [50–53]. For this, each experimental or simulated +system is divided up into blocks of size l = 1 mm. At +this scale the filament density is relatively homogeneous, +but the blocks are large enough to have good statistics. +The local order parameter S = ⟨cos(2ˆθ)⟩ is measured for +filament orientations ˆθ taken with respect to the local +nematic director (see supplemental materials [42] for de- +tails). We then calculate ⟨S⟩ as a block average, which is +capable of quantifying the emergence of local order even +in a globally heterogeneous system [52, 54]. +Both experiments and simulations show low nematic +order at low densities. +At higher ρ the appearance of +collective structures is captured by a sharp increase in +⟨S⟩, as shown in Fig. 3(i). Experimentally, the transition +from a disordered state, with ⟨S⟩ ≃ 0.2, to an ordered + +0.6 +exp. +fit +Probability p(Vo +0.0 +0 +2 +6 +Speed vo (μm s-11 +2 +T +314 +1 +T +4 +T +exp. +sim. +T +0 +0.4 +5-4 +0.8 +T +T +3-2 +T0.4 +80 +# of interactions +60 +40 +# int. +Palign +0 +20 +△0/0 +0.0 +0 +T +Incident angle 4 +10-1 +101 +103 +Block size l 2 (mm2) +10-3 +10-2 +10-1 +100 +� +S +� +(a) sim. +l 2 = v2 +0/(Dωτ) +10-1 +101 +103 +Block size l 2 (mm2) +(b) exp. +measured radii +0 +20 +40 +60 +80 +100 +Density ρ(mm−2) +0.0 +2.5 +5.0 +7.5 +10.0 +v0(Dωτ)−1/2 (mm) +0 +2 +4 +6 +8 +10 +Crossover scale l ∗ (mm) +(c) +l ∗ = v0(Dωτ)−1/2 +exp. +ρ = 69mm−2 +ρ = 76mm−2 +ρ = 83mm−2 +0.0 +0.5 +1.0 +1.5 +Dω (s−3) +×10 9 +(d) + +25 mm +(e) +(f) +(g) +(h) +FIG. 4. Emergence of large-scale patterning. Finite-size scaling of the block-average order parameter ⟨S⟩ was investigated in +(a) simulations and (b) experiments, by varying the block size l for the same data shown in Fig. 3. The power-law decay at +low density indicates a disordered, isotropic state. The emergence of structures at high density is marked by a plateau lasting +until l reaches the size of the emerging structures, which we term the crossover lengthscale l∗, after which a more rapid decay +is observed. (c) For different model parameters l∗ can be compared to the characteristic scale at which active and diffusive +terms balance, ℓ∗. Snapshots show the resulting patterns for some simulations with (d): ρ = 83 mm−2, Dω = 1.2 × 10−9 s−3, +τ = 480 s. (e): ρ = 76 mm−2, Dω = 7.5 × 10−10 s−3, τ = 1920 s. (f): ρ = 83 mm−2, Dω = 7.5 × 10−10 s−3, τ = 320 s. (g): +ρ = 69 mm−2, Dω = 1.7 × 10−9 s−3, τ = 480 s, and (h) for filaments growing naturally under typical incubation conditions. +The characteristic scales of the patterns are shown by red circles of radius ℓ∗. +state of ⟨S⟩ ≃ 0.7 is seen at a critical density of about ρ = +40–50 mm−2. The simulations show a similar response. +Varying the model parameters somewhat does not change +the qualitative nature of the ordering transition, but does +affect the critical value of ρ. +For a ‘gas’ of weakly-interacting filaments, can we +predict when their interactions will become important +enough to lead to collective behavior? +For simplicity, +consider straight filaments of density ρ moving at speed +v0. Randomly oriented filaments will present an average +cross-sectional length ¯L = ⟨L sin θ⟩ = 2L/π to each other. +As one filament advances, it will then encounter others +at an average frequency f = ¯Lρv0 = 2Lρv0/π. Experi- +mentally, most interactions have no effect, but a fraction +a result in alignment. The rate of filament ordering thus +scales as af. In contrast, aligned filaments can split up, +which we can assume happens randomly with some rate +b. Under these simplistic but representative assumptions, +interactions should become important when the rates of +filament alignment and breakup balance, af ≃ b, and +this cross-over condition defines a characteristic density +ρc = +πb +2aLv0 +. +(3) +Using experimental values of a = 0.04 and b = 0.007 s−1 +(see supplemental materials [42] for details) predicts that +ρc ∼ O(50) mm−2. A disordered gas of filaments would +be expected for densities ρ ≪ ρc, with ordered states +starting to appear at densities ρ ≈ ρc. This simple es- +timate agrees well with the density of the disorder-order +transition in cyanobacteria, shown in Fig. 3. +In a similar way, we can also rationalize the emer- +gent lengthscale of the reticulate pattern as a signature +of the balance struck between activity and fluctuations. +By nondimensionalizing Eqs. (1)–(2), the ratio between +the angular rate of change and diffusion defines a P´eclet +number for our system, Pe = v0/(ℓ√Dωτ), where ℓ is +some reference length, and Dω and τ are as in Eqs. (1)– +(2). By imposing a balance of the active and diffusive +terms, i.e. Pe = 1, and using Dω = v2 +0δκ2/τ we predict +a characteristic size ℓ∗ = v0(Dωτ)−1/2 = 1/δκ ≈ 5 mm. +To substantiate this prediction, we perform a scaling +analysis [51, 52] of how the block-averaged order pa- +rameter ⟨S⟩ depends on the block size l. Figure 4(a,b) +shows the results for the simulations and experiments. +At low densities we see the simple power-law decay +expected for a disordered system [52]. +However, for +ρ > 40 mm−2, ⟨S(l)⟩ develops two distinct regimes: a + +11 +12 +13 +14 +155 +plateau at low l, reflecting the local order of the bundles, +and a more rapid decay at large l. From the position +of the crossover between these responses we extract a +lengthscale l∗ (methods in supplemental materials [42]), +as reported in Fig. 4(c). The drop in ⟨S⟩ above l∗ is at- +tributed to bundles with different orientations appearing +within the same block. +In the simulations, we explore the dependence of l∗ on +the model parameters, by varying τ, Dω, and ρ. Steady- +state snapshots of some representative systems are shown +in Fig. 4(d-g). While the fine details of the simulated +patterns vary, these show that the crossover lengthscale +l∗ is always consistent with the radius of the emergent +structures. As summarized in Fig. 4(c), we find that this +feature size generally matches the characteristic length +ℓ∗ = 1/δκ predicted via the P´eclet number. Extending +this test to the experiments, the radius of the structures +of dense colonies was measured to be l∗ = 3.5 ± 0.6 mm, +which matches with ℓ∗ = 1/δκ = 2.9 mm. +A demon- +stration of this, for patterns in a typical bottle of our +cyanobacteria stock, is shown in Fig. 4(h). +In summary, we have studied colonies of filamentous +cyanobacteria and their emerging collective organization. +The length of the filaments is comparable to other scales +in the problem (e.g. +curvature) and can couple with +them; we thus cannot assume separation of scales. +A +theoretical model that accounts for fluctuations, large as- +pect ratios, motility, and nematic alignment reproduces +the structure of the reticulate pattern. Taken together, +our results point at a new class of active matter charac- +terized by the following features: (i) Elongated filaments +with position-dependent orientation and multiple inter- +action sites along each filament. +These details are re- +quired to understand the extended interactions and the +critical density at which the cyanobacteria condense into +collective structures like bundles; (ii) A gliding motility +induced by polar forces (see also Ref. [23]), unlike the +extensile/contractile nature of microtubule-kinesin sys- +tems [44] or confluent epithelia [55]; And (iii) the path- +tracking dynamics of the body following its head, sub- +ject to fluctuations and active motion, which are ulti- +mately responsible for the reticulate pattern, and which +select a well-defined lengthscale of that emergent pat- +tern. Cyanobacteria are an important class of microbial +life, and among the earliest form of multicellular organ- +isms. We note that the parameters governing their self- +organization identified here are evolutionarily selectable +traits, which can inform the study of the fossil record. +ACKNOWLEDGMENTS +The authors thank Maike Lorenz (SAG G¨ottingen) for +support on cyanobacteria cultures, Stefan Karpitschka +(MPIDS) and Jack Paget (Loughborough) for discussions +and Graham Hickman (NTU) for microscopy support. +Microscopy facilities were provided by the Imaging Suite +at the School of Science and Technology at Nottingham +(b) +(d) +(c) +(a) +Reversal +2 mm +1.5 mm +FIG. S1. Motion of isolated O. lutea. (a) Filaments followed +smoothly curving paths, as shown here by the track of one +midpoint over time. (b) The filaments occasionally reversed +their direction of motion, but their path curvature was main- +tained across such events, as in the track shown here. +(c) +The path curvature, κ, along any track fluctuated in time, +with negative values indicating clockwise motion; data here +are from the track in panel (a), before and after smoothing. +(d) The autocorrelation of the filtered data is well fit by an +exponential decay with a correlation time τ. The insert shows +the distribution of τ for different filaments. +Trent University. Numerical calculations were performed +using the Sulis Tier 2 HPC Platform funded by EPSRC +Grant EP/T022108/1 and the HPC Midlands+ consor- +tium. We gratefully acknowledge the use of the Lovelace +HPC service at Loughborough University. This work was +supported by the MPIDS. +SUPPLEMENTAL INFORMATION +EXPERIMENTAL METHODS +Culture preparation. +Stock of Oscillatoria lutea +(SAG 1459-3) was maintained in a medium of BG11 +broth (Sigma-Aldrich) diluted to a ratio of 1:100 with +deionized water. Following Lorentz et al. [56], samples +were incubated at 20 ± 1 ◦C, with warm-white LEDs +(color temperature of 2800 K) providing a photon flux +of 10 ± 2 µmol m−2 s−1 on a 16 h day + 8 h night cy- +cle. For sampling, material was transferred into a 100 +ml bottle half-filled with medium and shaken mildly to +separate the filaments. Samples were then drawn into a +syringe and added dropwise to 6-well plates (34 mm well +diameter) three-quarters filled with medium. The colony +density ρ was controlled by varying the number of drops +added to each well. The well plates were covered and left +in the incubator for 72 hours before imaging. +Imaging. +We used a confocal laser scanning mi- +croscope (Leica TCS SP5) in bright field and fluores- +cence modes. +Fluorescence of the chlorophyll-a in the + +80 +60 +40 +20 +0T150 +100 +50 +0ime6 +(a) +(c) +(d) +𝜌 = 2 mm-2 +𝜌 =35 mm-2 +(b) +fit - halfnormal +FIG. S2. Filament curvature changes with density ρ. (a) At low densities filaments are visibly curved in shape while (b) at +higher densities the filaments are straighter; scale bars are 1 mm. Filaments highlighted in green have been manually masked for +curvature measurements. (c) The average curvature drops with ρ, while the standard deviation (error bars) remains relatively +constant. (d) Histograms of |κ| show the bias towards a preferred curvature at low ρ. At higher ρ the curvature distribution is +consistent with a normal distribution centered around zero curvature. +cyanobacteria was excited by the 514 nm line of the ar- +gon laser at 29% power. The light emitted was detected +through a 620–780 nm band-pass filter by a HyD hy- +brid detector at 100% gain. Images were observed with +a PL Fluotar 10X/0.3na air objective with a pinhole of +70.8 µm. +Scanned image frames were 512×512 pixels +(1.55×1.55 mm2) with a scan rate of 400 Hz. No aver- +aging or integration was applied during collection. Dy- +namic measurements were made at fixed positions or with +manual tracking. Wide area imaging (e.g. Figs. 1, 3) +used the microscope’s tile scan protocol, and were recon- +structed from overlapping frames collected by the rapid +progressive scan of regions of interest. Images were bi- +narized in Matlab using the adaptthresh algorithm to +ensure a consistent appearance of individual filaments, +then skeletonized and despurred for further analysis. +PARAMETER MEASUREMENT +Isolated filaments. +The filaments had lengths of +L = 1.5 ± 0.5 mm, as measured along their skeleton, and +cross-sectional diameters of σ = 4.2 ± 0.2 µm, as mea- +sured in Ref. [41]. We followed the motion of 23 isolated +(ρ < 1 mm−2) filaments over time, generating time series +of the positions of the head, tail and midpoint of each +skeletonized filament over observation periods of up to 3 +hours. The distribution of instantaneous speeds, v0, in +Fig. 2(a) was measured using the midpoint tracks and a +40-point moving window. There was no significant cor- +relation of speed with filament length. +Figure S1(a) shows the isolated filaments tracing +smoothly curving paths, along which the curvature fluc- +tuates over time. Filamentous cyanobacteria can inter- +mittently reverse the direction of their motion [27, 57], +see Fig. S1(b). Such reversals can affect the collective be- +havior of filaments that are sufficiently confined so as to +prevent filament crossings [23], although this limit is far +from our experimental conditions. We observed 25 spon- +taneous reversals during 23 h of single-filament tracking, +at intervals between 10 minutes and several hours. Af- +ter a reversal, a filament typically continued along a new +path with a curvature close to its pre-reversal value. +Path curvatures, κ = dθ/ds, were quantified from the +time-lapse image sets. A tangent-line fit to the central +half of each filament was used to measure its orienta- +tion θ, and the path coordinate s was taken from the +track of its midpoint. A numerical derivative for dθ/ds +was then calculated using a 40-point moving window in +time. +To remove high-frequency noise, resulting from +the numerical differentiation, the curvature data was +smoothed by a third-order Savitzky-Golay filter (method +adapted from [26], demonstrated in Fig. S1(c)). +The +smoothed path curvature data are consistent with time +series of curvatures measured by fitting circular arcs to +the filament skeletons (methods adapted from [41]). The +path curvature autocorrelation function was calculated +as ⟨κ(t)κ(t + t′)⟩/⟨κ2⟩, for delay t′. For each of the 23 +filament tracks a correlation time τ was found by fitting +the exponential relaxation e−t′/τ to the autocorrelation +function, see Fig. S1(d). The distribution of τ for all fil- +aments studied had a mean of 470 s and standard devia- +tion of 290 s. To check the robustness of these methods, +correlation times of 540 ± 300 s were calculated in the +same way, but starting from the time series of filament +curvatures (i.e. as fit by circular arcs). +Interacting filaments. We quantified the pair-wise +interactions of gliding filaments in colonies with interme- +diate densities of ρ ≃ 10 mm−2. +Filaments interacted +through contact, when their paths crossed. 400 such in- +teractions were tracked; in each case the angle of inci- +dence was taken with respect to the forward directions +of motion of the filaments at the point of contact. As +summarized in Fig. 2(b), the incident filament either +turned to follow beside the filament it met, or the two +filaments crossed over/under each other without altering + +7 +ො𝑛 +(a) +(b) +(c) +l = 1 mm +1 mm +l +FIG. S3. Calculation of the experimental order parameter. (a) Images were skeletonized, with an orientation or direction +assigned to every pixel on the skeleton, and partitioned into blocks of size l; the legend shows the color-coding of the orientation. +(b) Locally, S was calculated from the distribution of orientations within any particular block, with respect to the local director, +ˆn. (c) A block average then gives the global order parameter, ⟨S⟩, which depends on the size of the blocks used, l. Error bars +give the standard deviation across blocks. For the analysis in Fig. 3 we use a representative block size of l = 1 mm (red line). +their paths. Only 16 events resulted in alignment, giving +this outcome a relative probability of a = 0.04. Of these, +there were 10 cases of parallel alignment, and 6 of anti- +parallel alignment, where the newly bundled filaments +moved in opposite directions along adjacent paths. +After aligning, filaments travel together for some time, +before one filament breaks off onto a separate path. After +each of the 16 alignment events, we tracked the distance +traveled before the pair broke up. The mean distance +traveled while being aligned was db = 430 µm with stan- +dard deviation 200 µm. A similar effect was seen at walls, +where filaments that hit a wall curved to follow it for an +average of 520 ± 280 µm, before breaking away (averaged +over 100 observations). A representative rate of filament +breakup, b = 0.007 s−1, was calculated as v0/db. +The motion of filaments in bundles was characterized +using time-lapse images from seven locations clustered +along a single long bundle. Directions of the motion of +505 filaments were measured by hand, in ImageJ [58]. +For each location, we defined a local average orientation, +maintaining a consistent sense of the motion along the +bundle (in this case, with angles near zero implying mo- +tion roughly from the top to the bottom of the image). +Figure 2(c) shows the relative directions of motion of the +filaments, measured with respect to their local nematic +director. +Finally, we observed the effects of interactions on fil- +ament shapes. +In isolation O. lutea filaments tend to +glide in clockwise rotation, with a preferred curvature +of 540 ± 230 m−1 [41]. +Here, we measured curvature +by manually masking individual filaments in thresholded +images of colonies at various densities, as shown in +Fig. S2(a,b). +The masked filaments were skeletonized, +despurred, and circular arcs were fit to their shapes. The +mean and standard deviation of the resulting distribu- +tions of unsigned curvatures are shown in Fig. S2(c). At +0 +0 +0 +1 +1 +1 +2 +2 +2 +3 +3 +3 +FIG. S4. +Sketch of the motion of modeled filaments. +For +simplicity, a filament of four beads (marked from α = 0 to +3, where 0 is the head) is shown at three generic time steps: +t−∆t, t, and t+∆t. At each time step, the tail bead (α = 3) is +removed and a new bead is placed in the front of the filament +as the new head. An arrow indicates the direction of motion. +low densities, Fig. S2(d–top), the results are similar to +isolated filaments. As their density increases, Fig. S2(d– +bottom), the filaments become straighter on average, and +the peak of the curvature distribution shifts towards zero. +As a best estimate of filament shapes in dense colonies, +we combined observations from ρ = 35, 41 and 49 mm−2, +and fit the results with a Gaussian distribution of zero +mean. Figure S2(d) shows the fit, which gives a repre- +sentative spread of curvatures of δκ = 340 ± 40 m−1. +Order parameter. The 2D nematic order parameter, +⟨S⟩, was calculated using the GTFiber App [51] via the +structure tensor method [51, 52]. This assigns an orien- +tation from 0 to π to each pixel on an image skeleton, +and then divides the image up into blocks of size l, as +in Fig. S3(a). In each block the local order parameter +S = ⟨cos(2θn)⟩ is calculated, where the orientation θn is +measured with respect to the local director, or average +orientation of pixels, within that block (see Fig. S3(b)). + +元/28 +100 +103 +10-2 +10-1 +100 +� +S +� +ρ = 83mm−2 +τ = 480s +D = 1.2 × 10−9 s−3 +100 +103 +ρ = 76mm−2 +τ = 1920s +D = 7.5 × 10−10 s−3 +100 +103 +Block size l 2 (mm2) +ρ = 83mm−2 +τ = 320s +D = 7.5 × 10−10 s−3 +100 +103 +ρ = 69mm−2 +τ = 480s +D = 1.7 × 10−9 s−3 +simulations +ensemble average +asymptotic fits +combined fit +estimated l ∗2 +FIG. S5. Examples of the procedure used to identify the scale of reticulate patterns. The order parameter ⟨S⟩ is calculated +within blocks of various size l, and averaged (black line) over five independent simulations (gray lines) for each set of parameters +ρ, τ, Dω. Linear fits identify the asymptotic slopes (red dashed lines), which are smoothly connected to estimate a crossover +length scale l∗ (green circle). The four examples shown correspond to the simulations in Fig. 4(d–g). +The global order parameter ⟨S⟩ is taken as the average +of S over all non-empty blocks. As shown in Fig. S3(c), +its value depends on the block size, l. +We characterized ⟨S⟩ in 50 colonies prepared identi- +cally (see culture preparation), but with different ρ. In +each case a 17 × 17 mm2 region of interest was cropped +from the center of a confocal image of the whole colony, to +minimize the influence of the chamber boundaries. The +data in Fig. 3 use a representative block size of l = 1 mm, +which is large enough to provide a good statistical aver- +age within each box, but small enough to still give a +homogeneous sampling. Results in Fig. 4 are prepared in +the same way, but with varying l. +NUMERICAL METHODS +Simulations. We used 72×72 mm2 domains with pe- +riodic boundary conditions to model the motion of N = +36 000–504 000 filaments, corresponding to the density +range ρ = 7–97 mm−2. Each filament i had a fixed speed +vi and was discretized into a chain of beads with posi- +tions ri,α, where the index α counts beads from the head, +α = 0, to the tail. At each time step, of size ∆t = 0.5 s, +the bead at the tail end of each filament is removed, and a +new bead is added as its new head. All other beads incre- +ment their index, α → α + 1, without changing position, +as sketched in Fig. S4. This economical move reproduces +the experimental behavior where the head of a filament +leads, while the rest of the filament follows in its track. +To determine the updated location and orientation of +the filament heads after each time step, we use the Euler– +Maruyama algorithm to solve Eqs. (1-2). If the head of +a filament is within the interaction range d of any links +between two beads of another filament, the interaction +potential U(θi, θj) is calculated based on the current ori- +entation of the head, θi, and the orientation of the closest +link, θj. All measurements are made after an equilibra- +tion time of 105 s of simulated time. +Determining crossover length. The scaling of the +block-average order parameter ⟨S⟩ with the block size l +can reveal structural information [51, 52]. We calculated +⟨S⟩ for simulations in the same way as the experiments +(see Fig. S3), based on the orientations of beads within +each block, and averaging over an ensemble of five in- +dependent simulations with different random seeds. At +higher densities there were two distinct scaling regimes. +As shown in Fig. S5, there is a crossover lengthscale be- +tween these limits, l∗ (green dot), which we use as an in- +dication of the size of the emergent structures. We found +that a robust way to identify l∗ was through the intercept +of the two asymptotic power laws. To this end, we per- +formed linear least-squares fits of log⟨S⟩ = n1 log(l2)+b1 +for l2 < 1 mm2 and log⟨S⟩ = n2 log(l2) + b2 for l2 > +100 mm2. To smoothly connect the two cases we then fit +log⟨S⟩ = (n2 − n1) log(l2 − l∗2) + n1 log(l2) + b +to the whole range of data, with l∗ and b as fitting pa- +rameters. The resulting fits are shown in Fig. S5 for some +different parameter choices of ρ, τ, and Dω. 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Methods +9(7), 676–682 (2012). + diff --git a/WtFJT4oBgHgl3EQf4y1Z/content/tmp_files/load_file.txt b/WtFJT4oBgHgl3EQf4y1Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..759f5b723f6771731b70e3e7a048074388272550 --- /dev/null +++ b/WtFJT4oBgHgl3EQf4y1Z/content/tmp_files/load_file.txt @@ -0,0 +1,913 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf,len=912 +page_content='Active Spaghetti: Collective Organization in Cyanobacteria Mixon K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Faluweki,1, 2, ∗ Jan Cammann,3, ∗ Marco G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Mazza,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' † and Lucas Goehring1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' ‡ 1School of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Nottingham Trent University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Nottingham NG11 8NS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' UK 2Malawi Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Malawi University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Limbe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Malawi 3Interdisciplinary Centre for Mathematical Modelling and Department of Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Loughborough University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Loughborough,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Leicestershire LE11 3TU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' United Kingdom 4Max Planck Institute for Dynamics and Self-Organization (MPIDS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Am Faßberg 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 37077 G¨ottingen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Germany (Dated: January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 2023) Filamentous cyanobacteria can show fascinating patterns of self-organization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' which however are not well-understood from a physical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' We investigate the motility and collective orga- nization of colonies of these simple multicellular lifeforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' As their area density increases, linear chains of cells gliding on a substrate show a transition from an isotropic distribution to bundles of filaments arranged in a reticulate pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Based on our experimental observations of individ- ual behavior and pairwise interactions, we introduce a model accounting for the filaments’ large aspect ratio, fluctuations in curvature, motility, and nematic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' This minimal model of active filaments recapitulates the observations, and rationalizes the appearance of a characteristic lengthscale in the system, based on the P´eclet number of the cyanobacteria filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Collective organization is a defining feature of living matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' It has received vivid attention [1–7] for its appli- cations in the life sciences [8, 9], and as an example of how nonequilibrium forces can drive sustained flows of matter and energy [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The first seminal studies of active matter treated the motion of point-like particles [1, 2, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Non-reciprocal interactions between even such simple ob- jects, with a single orientation, allow access to states not possible in equilibrium systems [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Long, flexible fila- ments, whose orientation varies along their length, offer the opportunity to study different classes of active matter [12–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Because there can be many points of interaction per filament, correlations can spread over long distances, opening the door to novel behavior [20, 24–29], whose complete understanding remains lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' As an important example of such a type of active mat- ter, cyanobacteria are among the most abundant organ- isms on Earth [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' They evolved the photosynthetic mechanisms that led to our oxygen-rich atmosphere and perform nearly all nitrogen fixation in marine environ- ments [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Filamentous cyanobacteria also straddle the boundary between single and multicellular organisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' they can grow into chains of cells several millimeters long through ‘filamentation’, perhaps the oldest form of mul- ticellularity [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Many species live on solid surfaces, including stromatolites [33, 35], and move by a gliding motility mechanism [36–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Dense colonies can develop complex structures, such as reticulate patterns [27, 39], over hours or days, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Despite their importance to the development of com- plex life on our planet, and their potential impact for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' carbon-neutral biofuels [40], no general mechanism has been identified to rationalize the collective behav- ior of filamentous cyanobacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Here, we demonstrate ∗ equal contribution † E-mail: m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='mazza@lboro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='uk ‡ E-mail: lucas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='goehring@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='uk that the emergent patterns of their colonies can be appre- hended as the collective result of independently moving actors with simple interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' By modeling these dy- namics we show how distinctive features of filamentous cyanobacteria, such as their large aspect ratio and the tendency of a filament to follow the trail laid down by its head, enable the accurate prediction of features like the critical density associated with the appearance of collec- tive ordering, and the emergent lengthscale of features above that density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' To this end, we investigate the active self-organization of Oscillatoria lutea, a typical strain of filamentous cyanobacteria, consisting of simple (non-branching, non- heterocystous) chains of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Details of our cultivation and measurement methods are provided as supplemental appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' In our cultures, the cyanobacteria filaments have well-defined widths σ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='2 µm [41] and lengths L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='5 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' In all cases here error ranges report the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' In relative isolation, with area densities ρ ≃ 1 mm−2, the filaments move at speeds v0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='7 µm s−1, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' They glide along smoothly curv- ing paths, which are characterized by tracking the ori- entation θ of the tangent to the midpoints of filaments through time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The curvature κ = dθ/ds of the path s(t) traced by any filament fluctuates slowly with time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' the auto-correlation of κ is well-described by an exponential relaxation with an autocorrelation time of τ = 470±290 s (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Isolated filaments are biased towards clock- wise motion, as in related species [36, 37, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' How- ever, from densities as low as ρ = 6 mm−2 and up to ρ = 49 mm−2 the filaments adopt straighter shapes on average (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' These distributions of curvatures peak around zero, with standard deviation of δκ = 340 ± 40 m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' To quantify the interactions between filaments, we ob- serve cases where the head (leading end) of one filament approaches and intersects another filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' In most of these pairwise interactions there is no direct effect, rather arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='11667v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='soft] 27 Jan 2023 2 30 𝜇m 10 mm (a) (b) (c) 500 𝜇m FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The structure of a colony of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' lutea at density ρ = 53 mm−2 shows (a) a reticulate pattern, with (b) the local alignment of filaments within bundles, and (c) filament motion (arrows) that is predominantly parallel or anti-parallel to neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' the filaments simply pass over/under each other with- out changing path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' However, in a few cases, 4% of the time, the incident filament is deflected, turning to travel alongside the other filament, which typically remains un- perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Aligning interactions only happen for small angles of incidence, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 2(b), and result in the two filaments moving parallel or anti-parallel, de- pending on the angle of approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' After aligning, the filaments track each other for some distance, on average 430 µm, before one starts to split away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' These interac- tions are fundamentally non-reciprocal [7], as the align- ment response is path-dependent [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The pairwise interactions promote the formation of bundles of aligned filaments, which can organize denser colonies into a higher-level architecture, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' We confirmed the local nematic nature of this or- dering by observing the motion of nearby filaments along one bundle, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' All filaments in the bundle are well-aligned, with approximately equal fractions (223 versus 282 filaments, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 2(c)) traveling in either direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Between the bundles is a dilute ‘gas’ of more randomly oriented filaments, similar in appearance to the disordered colonies seen at lower densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Some of these behaviors, such as nematic alignment and the tendency to form dynamic bundles and networks, are reminiscent of those of microtubules at an inter- face [9, 26, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' However, there are also conspicuous dif- ferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Most importantly, the length of a cyanobacteria filament is comparable to other characteristic lengths of this system, such as the filament’s radius of curvature, or the emergent pattern lengthscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Hence, there is no a priori clear separation of scales, and we will show that the elongated nature of the cyanobacteria filaments af- fects the nature of their collective self-organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' A benefit of this perspective is that it leads directly to a relatively simple model that can be informed in all its parameter choices by experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' We treat the cyanobacteria as motile one-dimensional objects discretized by chains of point-like beads (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 2(d)), as befits their large aspect ratio, L/σ > 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' For simplicity, all chains have length L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='5 mm, and representative disorder is introduced via their motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Their speeds are constant in time, but initially drawn from a normal distribution with average v0 = 3 µm s−1 and standard de- viation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='7 µm s−1, in accordance with experimental ob- servations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The position ri,α of bead α of chain i follows the track laid out by its head, so that ri,α(t) = ri,α−1(t − ∆t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' At each time step ∆t the end bead is removed from the tail of each chain, and a new bead is added at its head, displaced by a distance vi∆t at an angle θi (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Similar models have been applied to isolated filaments [22] and filaments moving on a lattice [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Although this system has some similarities to active polymers [13, 15–17, 46], those lack a unique curvature autocorrelation time and instead each portion of the polymer fluctuates independently;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' in contrast, the fluctuations and curvature of our chains are solely deter- mined by their heads and not by each segment thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Motivated by the models of active nematic particles used to simulate microtubules [26, 47], C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' elegans [48] and Pseudanabaena sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' [49], we now introduce a model of interacting active chains, as appropriate to the behavior of filamentous cyanobacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Here, the orientation θi and angular velocity ωi of the head of each chain i evolves according to a modified Ornstein–Uhlenbeck process dωi dt = −1 τ dθi dt + � 2Dωξi(t), (1) dθi dt = ωi − J Nij � j∼i ∂ ∂θi U(θi, θj), (2) where τ is the curvature autocorrelation time, Dω is the orientational diffusion coefficient, and ξi(t) is a Gaussian white noise with zero mean and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Dω is not directly accessible experimentally, but can be linked to other physical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Without any filament- filament interactions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (1) produces a normal distri- bution of angular velocities with zero mean and variance 3 (d) (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Filament behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (a) The distribution of experi- mentally observed gliding speeds (blue) is well-fit by a Gaus- sian (red, used for simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (b) Interacting filaments align with a probability that depends on their angle of inci- dence (histogram, left axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' interaction frequency, right axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' In the model, an incident chain is deflected on average by the relative angle ∆θ/θ (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (c) In bundles, the direc- tions of filament motion have a nematic distribution: a polar histogram compares experimental (blue) and simulated (red) cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (d) Schematic of a modeled pairwise interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' When the head of a chain is within distance d of another chain, it experiences an aligning effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' ⟨ω2⟩ = Dωτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' For chains moving at speed v0 this trans- lates into a curvature distribution with standard devi- ation δκ = � ⟨ω2⟩/v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Hence, Dω = (v0δκ)2/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Fi- nally, interactions of strength J are modeled by a ne- matic Lebwohl–Lasher potential, U = − cos [2(θi − θj)], averaged over the Nij chains within an interaction range d of the head of chain i, where θj is the orientation of the nearest bead on chain j (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 2(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The model parameters were matched to experimen- tal values of relatively isolated cyanobacteria, and fine- tuned based on the collective behavior at higher densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' In particular, unless otherwise stated, we set τ = 480 s, δκ = 200 m−1 (resulting in Dω = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='5 × 10−10 s−3) and d = 5 µm, close to the observed values of 470 s, 340 m−1 and the filament diameter of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='2 µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The interaction strength, J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='006 s−1, was chosen by con- sidering two filaments meeting at an angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' On average, the effects of interactions are comparable if the incident filament is either deflected by a relative angle ∆θ/θ, or by the whole angle θ with probability palign(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 2(b), in this sense J gives a similar average response to the experimentally observed interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Experimentally, colonies of cyanobacteria filaments are (g) (a) 10 mm (c) (f) (h) (d) (b) 0 20 40 60 80 Filament density ρ(mm−2) 0 1 � S � (i) sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Collective behavior and disorder-order transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Panels (a–d) show micrographs of colonies at densities ρ = 25, 31, 42, and 59 mm−2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Panels (e–h) show snapshots of simulations at comparable densities of ρ = 24, 31, 41 and 59 mm−2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' To avoid boundary ef- fects, simulations were performed in systems with sides 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='5× larger than shown;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' the panels are cropped to match the mi- crograph size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (i) Order parameter ⟨S⟩, averaged over 1 mm2 blocks covering the experimental (blue) or simulated (red) domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Error bars and shading give the standard deviation of ⟨S⟩ over the blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' At low ρ the filaments are randomly aligned, but locally-nematic bundles and a reticulated struc- ture emerge above ρ ∼ 40 mm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' disordered at low density, but show emergent patterns at higher densities, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 3(a–d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The simulated chains order in a similar way, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 3(e–h), with reticulated structures appearing at higher ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' We quantify local order in the steady-state by means of the 2D nematic order pa- rameter [50–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' For this, each experimental or simulated system is divided up into blocks of size l = 1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' At this scale the filament density is relatively homogeneous, but the blocks are large enough to have good statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The local order parameter S = ⟨cos(2ˆθ)⟩ is measured for filament orientations ˆθ taken with respect to the local nematic director (see supplemental materials [42] for de- tails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' We then calculate ⟨S⟩ as a block average, which is capable of quantifying the emergence of local order even in a globally heterogeneous system [52, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Both experiments and simulations show low nematic order at low densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' At higher ρ the appearance of collective structures is captured by a sharp increase in ⟨S⟩, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Experimentally, the transition from a disordered state, with ⟨S⟩ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='2, to an ordered 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='6 exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' fit Probability p(Vo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='0 0 2 6 Speed vo (μm s-11 2 T 314 1 T 4 T exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' T 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='4 5-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='8 T T 3-2 T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='4 80 # of interactions 60 40 # int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Palign 0 20 △0/0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='0 0 T Incident angle 4 10-1 101 103 Block size l 2 (mm2) 10-3 10-2 10-1 100 � S � (a) sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' l 2 = v2 0/(Dωτ) 10-1 101 103 Block size l 2 (mm2) (b) exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' measured radii 0 20 40 60 80 100 Density ρ(mm−2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='0 v0(Dωτ)−1/2 (mm) 0 2 4 6 8 10 Crossover scale l ∗ (mm) (c) l ∗ = v0(Dωτ)−1/2 exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' ρ = 69mm−2 ρ = 76mm−2 ρ = 83mm−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='5 Dω (s−3) ×10 9 (d) 25 mm (e) (f) (g) (h) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Emergence of large-scale patterning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Finite-size scaling of the block-average order parameter ⟨S⟩ was investigated in (a) simulations and (b) experiments, by varying the block size l for the same data shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The power-law decay at low density indicates a disordered, isotropic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The emergence of structures at high density is marked by a plateau lasting until l reaches the size of the emerging structures, which we term the crossover lengthscale l∗, after which a more rapid decay is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (c) For different model parameters l∗ can be compared to the characteristic scale at which active and diffusive terms balance, ℓ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Snapshots show the resulting patterns for some simulations with (d): ρ = 83 mm−2, Dω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='2 × 10−9 s−3, τ = 480 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (e): ρ = 76 mm−2, Dω = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='5 × 10−10 s−3, τ = 1920 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (f): ρ = 83 mm−2, Dω = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='5 × 10−10 s−3, τ = 320 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (g): ρ = 69 mm−2, Dω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='7 × 10−9 s−3, τ = 480 s, and (h) for filaments growing naturally under typical incubation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The characteristic scales of the patterns are shown by red circles of radius ℓ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' state of ⟨S⟩ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='7 is seen at a critical density of about ρ = 40–50 mm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The simulations show a similar response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Varying the model parameters somewhat does not change the qualitative nature of the ordering transition, but does affect the critical value of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' For a ‘gas’ of weakly-interacting filaments, can we predict when their interactions will become important enough to lead to collective behavior?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' For simplicity, consider straight filaments of density ρ moving at speed v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Randomly oriented filaments will present an average cross-sectional length ¯L = ⟨L sin θ⟩ = 2L/π to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' As one filament advances, it will then encounter others at an average frequency f = ¯Lρv0 = 2Lρv0/π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Experi- mentally, most interactions have no effect, but a fraction a result in alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The rate of filament ordering thus scales as af.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' In contrast, aligned filaments can split up, which we can assume happens randomly with some rate b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Under these simplistic but representative assumptions, interactions should become important when the rates of filament alignment and breakup balance, af ≃ b, and this cross-over condition defines a characteristic density ρc = πb 2aLv0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (3) Using experimental values of a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='04 and b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='007 s−1 (see supplemental materials [42] for details) predicts that ρc ∼ O(50) mm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' A disordered gas of filaments would be expected for densities ρ ≪ ρc, with ordered states starting to appear at densities ρ ≈ ρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' This simple es- timate agrees well with the density of the disorder-order transition in cyanobacteria, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' In a similar way, we can also rationalize the emer- gent lengthscale of the reticulate pattern as a signature of the balance struck between activity and fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' By nondimensionalizing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (1)–(2), the ratio between the angular rate of change and diffusion defines a P´eclet number for our system, Pe = v0/(ℓ√Dωτ), where ℓ is some reference length, and Dω and τ are as in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (1)– (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' By imposing a balance of the active and diffusive terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Pe = 1, and using Dω = v2 0δκ2/τ we predict a characteristic size ℓ∗ = v0(Dωτ)−1/2 = 1/δκ ≈ 5 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' To substantiate this prediction, we perform a scaling analysis [51, 52] of how the block-averaged order pa- rameter ⟨S⟩ depends on the block size l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Figure 4(a,b) shows the results for the simulations and experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' At low densities we see the simple power-law decay expected for a disordered system [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' However, for ρ > 40 mm−2, ⟨S(l)⟩ develops two distinct regimes: a 11 12 13 14 155 plateau at low l, reflecting the local order of the bundles, and a more rapid decay at large l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' From the position of the crossover between these responses we extract a lengthscale l∗ (methods in supplemental materials [42]), as reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The drop in ⟨S⟩ above l∗ is at- tributed to bundles with different orientations appearing within the same block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' In the simulations, we explore the dependence of l∗ on the model parameters, by varying τ, Dω, and ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Steady- state snapshots of some representative systems are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 4(d-g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' While the fine details of the simulated patterns vary, these show that the crossover lengthscale l∗ is always consistent with the radius of the emergent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' As summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 4(c), we find that this feature size generally matches the characteristic length ℓ∗ = 1/δκ predicted via the P´eclet number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Extending this test to the experiments, the radius of the structures of dense colonies was measured to be l∗ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='6 mm, which matches with ℓ∗ = 1/δκ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='9 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' A demon- stration of this, for patterns in a typical bottle of our cyanobacteria stock, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 4(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' In summary, we have studied colonies of filamentous cyanobacteria and their emerging collective organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The length of the filaments is comparable to other scales in the problem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' curvature) and can couple with them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' we thus cannot assume separation of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' A theoretical model that accounts for fluctuations, large as- pect ratios, motility, and nematic alignment reproduces the structure of the reticulate pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Taken together, our results point at a new class of active matter charac- terized by the following features: (i) Elongated filaments with position-dependent orientation and multiple inter- action sites along each filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' These details are re- quired to understand the extended interactions and the critical density at which the cyanobacteria condense into collective structures like bundles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (ii) A gliding motility induced by polar forces (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' [23]), unlike the extensile/contractile nature of microtubule-kinesin sys- tems [44] or confluent epithelia [55];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' And (iii) the path- tracking dynamics of the body following its head, sub- ject to fluctuations and active motion, which are ulti- mately responsible for the reticulate pattern, and which select a well-defined lengthscale of that emergent pat- tern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Cyanobacteria are an important class of microbial life, and among the earliest form of multicellular organ- isms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' We note that the parameters governing their self- organization identified here are evolutionarily selectable traits, which can inform the study of the fossil record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank Maike Lorenz (SAG G¨ottingen) for support on cyanobacteria cultures, Stefan Karpitschka (MPIDS) and Jack Paget (Loughborough) for discussions and Graham Hickman (NTU) for microscopy support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Microscopy facilities were provided by the Imaging Suite at the School of Science and Technology at Nottingham (b) (d) (c) (a) Reversal 2 mm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='5 mm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Motion of isolated O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' lutea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (a) Filaments followed smoothly curving paths, as shown here by the track of one midpoint over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (b) The filaments occasionally reversed their direction of motion, but their path curvature was main- tained across such events, as in the track shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (c) The path curvature, κ, along any track fluctuated in time, with negative values indicating clockwise motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' data here are from the track in panel (a), before and after smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (d) The autocorrelation of the filtered data is well fit by an exponential decay with a correlation time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The insert shows the distribution of τ for different filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Trent University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Numerical calculations were performed using the Sulis Tier 2 HPC Platform funded by EPSRC Grant EP/T022108/1 and the HPC Midlands+ consor- tium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' We gratefully acknowledge the use of the Lovelace HPC service at Loughborough University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' This work was supported by the MPIDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' SUPPLEMENTAL INFORMATION EXPERIMENTAL METHODS Culture preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Stock of Oscillatoria lutea (SAG 1459-3) was maintained in a medium of BG11 broth (Sigma-Aldrich) diluted to a ratio of 1:100 with deionized water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Following Lorentz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' [56], samples were incubated at 20 ± 1 ◦C, with warm-white LEDs (color temperature of 2800 K) providing a photon flux of 10 ± 2 µmol m−2 s−1 on a 16 h day + 8 h night cy- cle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' For sampling, material was transferred into a 100 ml bottle half-filled with medium and shaken mildly to separate the filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Samples were then drawn into a syringe and added dropwise to 6-well plates (34 mm well diameter) three-quarters filled with medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The colony density ρ was controlled by varying the number of drops added to each well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The well plates were covered and left in the incubator for 72 hours before imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' We used a confocal laser scanning mi- croscope (Leica TCS SP5) in bright field and fluores- cence modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Fluorescence of the chlorophyll-a in the 80 60 40 20 0T150 100 50 0ime6 (a) (c) (d) 𝜌 = 2 mm-2 𝜌 =35 mm-2 (b) fit - halfnormal FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Filament curvature changes with density ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (a) At low densities filaments are visibly curved in shape while (b) at higher densities the filaments are straighter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' scale bars are 1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Filaments highlighted in green have been manually masked for curvature measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (c) The average curvature drops with ρ, while the standard deviation (error bars) remains relatively constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (d) Histograms of |κ| show the bias towards a preferred curvature at low ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' At higher ρ the curvature distribution is consistent with a normal distribution centered around zero curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' cyanobacteria was excited by the 514 nm line of the ar- gon laser at 29% power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The light emitted was detected through a 620–780 nm band-pass filter by a HyD hy- brid detector at 100% gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Images were observed with a PL Fluotar 10X/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='3na air objective with a pinhole of 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='8 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Scanned image frames were 512×512 pixels (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='55×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='55 mm2) with a scan rate of 400 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' No aver- aging or integration was applied during collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Dy- namic measurements were made at fixed positions or with manual tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Wide area imaging (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 1, 3) used the microscope’s tile scan protocol, and were recon- structed from overlapping frames collected by the rapid progressive scan of regions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Images were bi- narized in Matlab using the adaptthresh algorithm to ensure a consistent appearance of individual filaments, then skeletonized and despurred for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' PARAMETER MEASUREMENT Isolated filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The filaments had lengths of L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='5 mm, as measured along their skeleton, and cross-sectional diameters of σ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='2 µm, as mea- sured in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' We followed the motion of 23 isolated (ρ < 1 mm−2) filaments over time, generating time series of the positions of the head, tail and midpoint of each skeletonized filament over observation periods of up to 3 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The distribution of instantaneous speeds, v0, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 2(a) was measured using the midpoint tracks and a 40-point moving window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' There was no significant cor- relation of speed with filament length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Figure S1(a) shows the isolated filaments tracing smoothly curving paths, along which the curvature fluc- tuates over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Filamentous cyanobacteria can inter- mittently reverse the direction of their motion [27, 57], see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Such reversals can affect the collective be- havior of filaments that are sufficiently confined so as to prevent filament crossings [23], although this limit is far from our experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' We observed 25 spon- taneous reversals during 23 h of single-filament tracking, at intervals between 10 minutes and several hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Af- ter a reversal, a filament typically continued along a new path with a curvature close to its pre-reversal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Path curvatures, κ = dθ/ds, were quantified from the time-lapse image sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' A tangent-line fit to the central half of each filament was used to measure its orienta- tion θ, and the path coordinate s was taken from the track of its midpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' A numerical derivative for dθ/ds was then calculated using a 40-point moving window in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' To remove high-frequency noise, resulting from the numerical differentiation, the curvature data was smoothed by a third-order Savitzky-Golay filter (method adapted from [26], demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The smoothed path curvature data are consistent with time series of curvatures measured by fitting circular arcs to the filament skeletons (methods adapted from [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The path curvature autocorrelation function was calculated as ⟨κ(t)κ(t + t′)⟩/⟨κ2⟩, for delay t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' For each of the 23 filament tracks a correlation time τ was found by fitting the exponential relaxation e−t′/τ to the autocorrelation function, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The distribution of τ for all fil- aments studied had a mean of 470 s and standard devia- tion of 290 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' To check the robustness of these methods, correlation times of 540 ± 300 s were calculated in the same way, but starting from the time series of filament curvatures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' as fit by circular arcs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Interacting filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' We quantified the pair-wise interactions of gliding filaments in colonies with interme- diate densities of ρ ≃ 10 mm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Filaments interacted through contact, when their paths crossed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 400 such in- teractions were tracked;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' in each case the angle of inci- dence was taken with respect to the forward directions of motion of the filaments at the point of contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' As summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 2(b), the incident filament either turned to follow beside the filament it met, or the two filaments crossed over/under each other without altering 7 ො𝑛 (a) (b) (c) l = 1 mm 1 mm l FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Calculation of the experimental order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (a) Images were skeletonized, with an orientation or direction assigned to every pixel on the skeleton, and partitioned into blocks of size l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' the legend shows the color-coding of the orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (b) Locally, S was calculated from the distribution of orientations within any particular block, with respect to the local director, ˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (c) A block average then gives the global order parameter, ⟨S⟩, which depends on the size of the blocks used, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Error bars give the standard deviation across blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' For the analysis in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 3 we use a representative block size of l = 1 mm (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' their paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Only 16 events resulted in alignment, giving this outcome a relative probability of a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Of these, there were 10 cases of parallel alignment, and 6 of anti- parallel alignment, where the newly bundled filaments moved in opposite directions along adjacent paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' After aligning, filaments travel together for some time, before one filament breaks off onto a separate path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' After each of the 16 alignment events, we tracked the distance traveled before the pair broke up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The mean distance traveled while being aligned was db = 430 µm with stan- dard deviation 200 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' A similar effect was seen at walls, where filaments that hit a wall curved to follow it for an average of 520 ± 280 µm, before breaking away (averaged over 100 observations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' A representative rate of filament breakup, b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='007 s−1, was calculated as v0/db.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The motion of filaments in bundles was characterized using time-lapse images from seven locations clustered along a single long bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Directions of the motion of 505 filaments were measured by hand, in ImageJ [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' For each location, we defined a local average orientation, maintaining a consistent sense of the motion along the bundle (in this case, with angles near zero implying mo- tion roughly from the top to the bottom of the image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Figure 2(c) shows the relative directions of motion of the filaments, measured with respect to their local nematic director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Finally, we observed the effects of interactions on fil- ament shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' In isolation O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' lutea filaments tend to glide in clockwise rotation, with a preferred curvature of 540 ± 230 m−1 [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Here, we measured curvature by manually masking individual filaments in thresholded images of colonies at various densities, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S2(a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The masked filaments were skeletonized, despurred, and circular arcs were fit to their shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The mean and standard deviation of the resulting distribu- tions of unsigned curvatures are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' At 0 0 0 1 1 1 2 2 2 3 3 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Sketch of the motion of modeled filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' For simplicity, a filament of four beads (marked from α = 0 to 3, where 0 is the head) is shown at three generic time steps: t−∆t, t, and t+∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' At each time step, the tail bead (α = 3) is removed and a new bead is placed in the front of the filament as the new head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' An arrow indicates the direction of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' low densities, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S2(d–top), the results are similar to isolated filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' As their density increases, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S2(d– bottom), the filaments become straighter on average, and the peak of the curvature distribution shifts towards zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' As a best estimate of filament shapes in dense colonies, we combined observations from ρ = 35, 41 and 49 mm−2, and fit the results with a Gaussian distribution of zero mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Figure S2(d) shows the fit, which gives a repre- sentative spread of curvatures of δκ = 340 ± 40 m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The 2D nematic order parameter, ⟨S⟩, was calculated using the GTFiber App [51] via the structure tensor method [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' This assigns an orien- tation from 0 to π to each pixel on an image skeleton, and then divides the image up into blocks of size l, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' In each block the local order parameter S = ⟨cos(2θn)⟩ is calculated, where the orientation θn is measured with respect to the local director, or average orientation of pixels, within that block (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 元/28 100 103 10-2 10-1 100 � S � ρ = 83mm−2 τ = 480s D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='2 × 10−9 s−3 100 103 ρ = 76mm−2 τ = 1920s D = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='5 × 10−10 s−3 100 103 Block size l 2 (mm2) ρ = 83mm−2 τ = 320s D = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='5 × 10−10 s−3 100 103 ρ = 69mm−2 τ = 480s D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='7 × 10−9 s−3 simulations ensemble average asymptotic fits combined fit estimated l ∗2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Examples of the procedure used to identify the scale of reticulate patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The order parameter ⟨S⟩ is calculated within blocks of various size l, and averaged (black line) over five independent simulations (gray lines) for each set of parameters ρ, τ, Dω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Linear fits identify the asymptotic slopes (red dashed lines), which are smoothly connected to estimate a crossover length scale l∗ (green circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The four examples shown correspond to the simulations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 4(d–g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The global order parameter ⟨S⟩ is taken as the average of S over all non-empty blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S3(c), its value depends on the block size, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' We characterized ⟨S⟩ in 50 colonies prepared identi- cally (see culture preparation), but with different ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' In each case a 17 × 17 mm2 region of interest was cropped from the center of a confocal image of the whole colony, to minimize the influence of the chamber boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 3 use a representative block size of l = 1 mm, which is large enough to provide a good statistical aver- age within each box, but small enough to still give a homogeneous sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 4 are prepared in the same way, but with varying l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' NUMERICAL METHODS Simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' We used 72×72 mm2 domains with pe- riodic boundary conditions to model the motion of N = 36 000–504 000 filaments, corresponding to the density range ρ = 7–97 mm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Each filament i had a fixed speed vi and was discretized into a chain of beads with posi- tions ri,α, where the index α counts beads from the head, α = 0, to the tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' At each time step, of size ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content='5 s, the bead at the tail end of each filament is removed, and a new bead is added as its new head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' All other beads incre- ment their index, α → α + 1, without changing position, as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' This economical move reproduces the experimental behavior where the head of a filament leads, while the rest of the filament follows in its track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' To determine the updated location and orientation of the filament heads after each time step, we use the Euler– Maruyama algorithm to solve Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' (1-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' If the head of a filament is within the interaction range d of any links between two beads of another filament, the interaction potential U(θi, θj) is calculated based on the current ori- entation of the head, θi, and the orientation of the closest link, θj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' All measurements are made after an equilibra- tion time of 105 s of simulated time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Determining crossover length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The scaling of the block-average order parameter ⟨S⟩ with the block size l can reveal structural information [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' We calculated ⟨S⟩ for simulations in the same way as the experiments (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S3), based on the orientations of beads within each block, and averaging over an ensemble of five in- dependent simulations with different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' At higher densities there were two distinct scaling regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S5, there is a crossover lengthscale be- tween these limits, l∗ (green dot), which we use as an in- dication of the size of the emergent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' We found that a robust way to identify l∗ was through the intercept of the two asymptotic power laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' To this end, we per- formed linear least-squares fits of log⟨S⟩ = n1 log(l2)+b1 for l2 < 1 mm2 and log⟨S⟩ = n2 log(l2) + b2 for l2 > 100 mm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' To smoothly connect the two cases we then fit log⟨S⟩ = (n2 − n1) log(l2 − l∗2) + n1 log(l2) + b to the whole range of data, with l∗ and b as fitting pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' The resulting fits are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' S5 for some different parameter choices of ρ, τ, and Dω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 4(d–g), l∗ gives a good estimate of the average radius of the emergent structures of the reticulated patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Vicsek, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Czir´ok, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Ben-Jacob, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Cohen, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Shochet, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} +page_content=' 75, 1226 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFJT4oBgHgl3EQf4y1Z/content/2301.11667v1.pdf'} 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b/X9AzT4oBgHgl3EQfKfvG/content/tmp_files/2301.01099v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9b7063983c5e4faaee6d7a10cf506872772f0300 --- /dev/null +++ b/X9AzT4oBgHgl3EQfKfvG/content/tmp_files/2301.01099v1.pdf.txt @@ -0,0 +1,1289 @@ +arXiv:2301.01099v1 [math.AG] 3 Jan 2023 +LOGARITHMIC MOTIVES WITH COMPACT SUPPORT +Nikolai Opdan +Abstract +We develop a theory of motives with compact support for logarithmic schemes over +a field. Starting from the notion of finite logarithmic correspondences with compact +support, we define the logarithmic motive with compact support analogous to the +classical case for schemes. +We then establish an analog of a Gysin sequence and, +assuming resolution of singularities, a Künneth formula. This implies that our theory is +□-invariant, which presents a critical feature that is absent in the classical case. Further +assuming resolution of singularities, we prove a duality theorem for log schemes which +we use to establish a cancellation theorem for log schemes whose underlying scheme is +proper. Moreover, we discuss new homology and cohomology theories for log smooth +fs logarithmic schemes based on our results. +1. +Introduction +A theory of motives for logarithmic schemes has recently been developed by Binda–Park– +Østvær [BPØ22a], [BPØ22b]. One of their motivating ideas is to develop a theory of motives +that captures information about a larger class of cohomology theories than those represent- +able in Voevodsky’s triangulated category of motives DMeff(k, Λ). Such theories include +crystalline cohomology, Hodge cohomology, and de Rham–Witt cohomology [BPØ22b]. Des- +pite not being A1-invariant, they satisfy a blow-up formula, projective bundle formula, Gysin +sequence, and Mayer-Vietoris, which indicates that one should create a more general linear +tensor category that also captures these theories. Extending the theories to log schemes, +one finds that they are insensitive to the log scheme +□ := (P1, ∞), +where the underlying scheme is the projective line and log structure given by the inclusion +of the divisor ∞ ֒→ P1. This indicates that one should create a theory of motives where □ +takes the place of the unit object A1. However, since □ is a log scheme, one needs to develop +a full motivic theory for log schemes to get a satisfactory framework. Thus starts our quest +for proving the fundamental properties of the triangulated category of logarithmic motives +logDMeff(k, Λ). +As a start, one expects to find analogs of classical results in this framework. Indeed, +Binda–Park–Østvær establishes a Gysin sequence, blow-up formula, projective bundle for- +mula, and a Thom isomorphism for log schemes; equipping schemes with a trivial log struc- +ture, this recovers the classical results in algebraic geometry. Assuming resolutions of singu- +larities, they moreover construct a fully faithful functor from DMeff(k, Λ) to logDMeff(k, Λ) +and identify its essential image. They also represent Hodge cohomology which shows that +the functor is not essentially surjective. These results enable one to view the new theory as +an enlargement of the classical theory for log schemes. +1 + +Nikolai Opdan +Voevodsky introduced in [Voe00] motives with compact support to state many essential +results such as the localization sequence ([Voe00, 4.1.5]) and the duality theorem ([FV00, +8.2]) for non-proper schemes. Importantly, they also describe dual objects ([Voe00, 4.3.2]) +and represent Bloch’s higher Chow groups for singular schemes ([Voe00, 4.2.9]). The goal +of this paper is to generalize this theory to logarithmic schemes. +We begin by defining logarithmic correspondences with compact support for a perfect +field k, and use this to define a presheaf with log transfers given by +Y �→ Λc +ltr(X)(Y ) := lCorc +k(Y, X) ⊗Z Λ ∈ Pshltr(k, Λ) +for any fs log scheme X and commutative ring Λ. After establishing its basic properties, +we show that its dividing Nisnevich sheafification a∗ +dNisΛc +ltr(X) is a dividing Nisnevich sheaf +with log transfers. Its image in logDMeff(k, Λ) we call the logarithmic motive with compact +support and we denote it by M c(X). It is contravariant for open embeddings and covariant +for closed embeddings. This construction resembles the classical construction of the motive +with compact support. +We show in Proposition 4.2 that M c(X) satisfies a Mayer–Vietoris property for coverings, +that is, for every strict Nisnevich square in lSm/k +Y ′ +Y +X′ +X +there is a homotopy cartesian square of log motives with compact support +M c(Y ) +M c(Y ′) +M c(X) +M c(X′). +Moreover, for any log modification f : X → Y in lSm/k we establish an isomorphism +M c(f) : M c(X) +≃ +−→ M c(Y ) +in Proposition 4.2. +The classical theory of motives benefits greatly from the localization sequence [Voe00, +4.1.5]. In our framework, there can be no such sequence since it would imply A1-invariance, +which is false since the essential image of the functor +Rω∗ : DMeff(k, Λ) → logDMeff(k, Λ) +consists of A1-local objects [BPØ22b, 8.2.16], but is not essentially surjective since Ωj +X/k is +an object of logDMeff(k, Λ) for every X ∈ SmlSm/k [BPØ22b, 9.7.1] and the groups +Hi +Zar(X, Ωj +X/k) ̸≃ Hi +Zar(X × A1, Ωj +X/k) +are not isomorphic. We instead provide a distinguished triangle that will take play the role +of the localization sequence; that is, we prove a slight generalization of the following (see +Theorem 4.3): +2 + +Logarithmic motives with compact support +– If Z is a smooth irreducible divisor on a smooth scheme X, there is a distinguished +triangle +M c(X, Z) → M c(X) → M c(Z)(1)[2] → M c(X, Z)[1] +in logDMeff(k, Λ). +Assuming resolution of singularities, an important consequence is that the logarithmic +motive with compact support factors over products, see Theorem 4.12. +– (Künneth formula) For every X, Y ∈ lSm/k there is an canonical map +M c(X) ⊗ M c(Y ) −→ M c(X × Y ) +which is an equivalence. +As a corollary we get the logarithmic motive with compact support is □-invariant, i.e., +– (Homotopy invariance) For every X ∈ lSm/k there is an isomorphism +M c(X × □) ≃ M c(X). +This shows an important feature that differs from the classical theory since the classical +motive with compact support is not homotopy invariant (A1-invariant) since M c(A1) ≃ +Λ(1)[2] by [Voe00, 4.1.8]. +Having settled basic properties, we continue to work under the assumption of resolution +of singularities, and relate our theory to the classical setting by establishing the isomorphism +HomlogDMeff(k,Λ)(M(Y )[n], M c(X)) ≃ HomDMeff(k,Λ)(M(Y − ∂Y )[n], M c(X)) +in Theorem 4.9. +This turns out to be helpful in proving a logarithmic analog of the duality theorem +[FV00, 8.2] in Theorem 4.15. +– (Duality) For every X ∈ Sm/k and Y, T ∈ lSm/k such that T is of pure dimension d +over k, there is an isomorphism +Hom(M(Y × T )[n], M c(X)) ≃ Hom(M(Y × T )(d)[2d + n], M c(X × (T − ∂T ))). +We then prove of a logarithmic analog of the cancellation theorem [Voe10] for log schemes +whose underlying scheme is proper in Theorem 4.16. +– (Cancellation) For every X, Y ∈ lSm/k such that the underlying schemes X, Y are +proper over k, there is an isomorphism +Hom(M(X), M(Y )) ≃ Hom(M(X)(1), M(Y )(1)) +induced by tensoring with Λ(1). +It remains an open problem to prove the cancellation +theorem for all fs log schemes. +We end our discussion in section 5 by defining new homotopy invariants for log smooth +fs log schemes which we call logarithmic cohomology with compact support defined by +Hn,i +lc (X, Λ) := HomlogDMeff(k,Λ)(M c(X), Λ(i)[n]) +and logarithmic Borel–Moore homology defined by +HlBM +n,i +(X, Λ) := HomlogDMeff(k,Λ)(Λ(i)[n], M c(X)). +Based on our results, we show that they are contravariant for open immersions, covariant +for closed immersions, has a Gysin triangle, and satisfies homotopy invariance and a duality +statement (under the assumption of resolution of singularities). +3 + +Nikolai Opdan +Outline – The paper is organized as follows: In section 2, we review the classical theory of +motives with compact support, and we briefly discuss the logarithmic motivic theory from +[BPØ22b] that will be used later. We begin our theory in section 3 with a discussion on +special correspondences needed to define the logarithmic motive with compact support in +section 4. This section presents the main body of the paper and is where we prove our main +results. Then, in section 5 we introduce new homology and cohomology theories for log +schemes and prove some of their fundamental properties. +Notation and conventions – Fix a perfect field k, Λ an unital commutative ring, and +let Sm/k denote the category of smooth and separated schemes of finite type over k and +Psh(Sm/k, Λ) the category of presheaves of Λ-modules on Sm/k with coefficients in Λ. By +convention, all log schemes are separated and of finite type over Spec k, where Spec k is the +point equipped with the trivial log structure. The category of all fs log schemes we denote +by lSch/k, while the category of all fs log smooth log schemes over k we denote by lSm/k. +For a log scheme X we let X denote the underlying scheme, and SmlSm/k denote the +full subcategory of lSm/k whose underlying scheme X is smooth over k. For a log scheme +X we let ∂X denote the set of points on X with non-trivial log structure. If X is an fs log +scheme, the complement of the log structure X −∂X will be an open subset of X, and there +is a canonical open immersion X − ∂X → X. +We often assume that the field k admits (strong) resolution of singularities. By this, we +mean that: +• for every integral scheme X of finite type over k, there exists a proper birational +morphism Y → X over k such that Y is smooth. +• let f : Y → X be a proper birational morphism of integral schemes over k where X +is smooth, and suppose that +f −1(X − (Z1 ∪ . . . ∪ Zr)) → X − (Z1 ∪ . . . ∪ Zr) +is an isomorphism. Then there is a sequence of blow-ups +Xn +fn−1 +→ Xn−1 +fn−2 +→ · · · +f0 +→ X0 = X +along smooth centers Wi such that the composition Xn → X factors through f, the +Wi are all contained in the preimage of Z1 ∪ . . . ∪ Zr in Xi, and the Wi have strict +normal crossing with the sum of the strict transforms of +Z1, . . . , Zr, f −1 +0 (W0), . . . , f −1 +i−1(Wi−1). +Especially, this is satisfied if k has characteristic 0 by the work of Hironaka [Hir64] (see also +[ATW19]). +Acknowledgments – I am grateful to my Ph.D. advisor Paul Arne Østvær for his con- +tinuous support and encouragement. I wish to thank Doosung Park, Alberto Merici and +Federico Binda for many helpful suggestions and fruitful discussions. +The research is supported by the RCN Frontier Research Group project no. 312472 +“Equations in Motivic homotopy”, and I gratefully acknowledge the hospitality of the Uni- +versità Degli Studi Di Milano and Centre for Advanced Study at the Norwegian Academy +of Science and Letters while writing this paper. +4 + +Logarithmic motives with compact support +2. +Background +2.1. +Motives with compact support – Voevodsky developed a theory of motives with +compact support in [Voe00]. The basic idea is that there should be a theory that accompanies +the theory of motives in the same way that cohomology with compact support accompanies +ordinary cohomology theories. Among other things, it was needed to state the localization +theorem ([Voe00, 4.1.5]) and the duality theorem ([FV00, 8.2]) for non-proper schemes, and +importantly it provided a description of the dualizing object, which for a smooth scheme X +takes the form +M(X)∗ := RHom(M(X), Λ(d))(−d) ≃ M c(X)(−d)[−2d] +in DMgm(k, Λ) ([Voe00, 4.3.7]). +Let Cork denote the category of finite correspondences, i.e., the objects are smooth +schemes over k, and morphisms from X to Y are finite correspondences. Recall the definition +of the motive M(X) in DMeff(k, Λ) for a smooth scheme X over k as the image of the +representable Nisnevish sheaf with transfers +Λtr(−) : Cork → Shvtr +Nis(Cork) +in DMeff(k, Λ). For any U ∈ Sm/k, one defines Λtr(X)(U) as the free abelian group with +coefficients in Λ of algebraic cycles on U ×X that are surjective and finite over a component +of U. One defines similarly the motive with compact support M c(X) for any scheme X of +finite type over k as the image in DMeff(k, Λ) of the sheaf with transfers +zequi(−, r) : Cork → Shvtr +Nis(Cork). +Here, for any for any U ∈ Sm/k, the group zequi(X, r)(U) is the free Λ-module of algebraic +cycles on U ×X that are dominant and of relative dimension r over a component of U [SV00]. +It can be checked that it is a Nisnevich sheaf with transfers, and that there is a natural +inclusion Λtr(X) ֒→ zequi(X, 0) which is an equivalence if X is proper. The Nisnevich sheaf +with transfers zequi(−, r) is covariant for closed maps and contravariant for étale morphisms +in T . In general, if f : Y → X is a flat equidimensional morphism of relative dimension n, +the flat pullback of cycles gives a morphism +f ∗ : zequi(X, r) → zequi(Y, r + n). +Similarly, if g : Z → X is a proper equidimensional morphism of relative dimension n, the +proper pushforward of cycles gives a morphism +g∗ : zequi(Z, r) → zequi(X, n + r). +For any commutative ring Λ one defines motivic cohomology with compact support with +coefficients in R by +Hn,i +c (X, Λ) := HomDMeff(k,Λ)(M c(X), Λ(i)[n], +and dually, one defines Borel–Moore motivic homology by +HBM +n,i (X, Λ) := HomDMeff(k,Λ)(Λ(i)[n], M c(X)). +5 + +Nikolai Opdan +For smooth schemes X, the fundamental comparison theorem ([Voe00, 4.2.9]) gives an iso- +morphism +Hn,i(X, Z) +≃ +−→ CHi(X, 2i − n) +between motivic cohomology and Bloch’s higher Chow groups. For non-smooth schemes Y +of dimension d, one instead have the isomorphism +CHi(Y, n) ≃ HBM +2(d−i)+n,d−i(Y, Z), +identifying higher Chow groups with Borel–Moore motivic homology groups. +To summarize, the motive with compact support has the following properties: +Properties 2.1. +i. (Contravariant functoriality) For any flat map X → Y of relative +dimension r there is a map +f ∗ : M c(Y )(r)[2r] → M c(X). +ii. (Covariant functoriality) For any proper map of relative dimension 0 g : X → Y there +is a map +g∗ : M c(X) → M c(Z). +iii. For every scheme X there is a canonical map +M(X) → M c(X), +which is an equivalence if X is proper over k. +iv. (Localization) Assume that k admits resolution of singularities. If Z +i→ X is a closed +subscheme with open complement U +j→ X, then there is a distinguished triangle +M c(Z) → M c(X) → M c(U) → M c(Z)[1] +(2.1) +([Voe00, 4.1.5]). +v. (Künneth formula) Assume that k admits resolution of singularities or the charac- +teristic is invertible in the coefficients. Then for any schemes X and Y there is an +isomorphism +M c(X × Y ) ≃ M c(X) ⊗tr +L,Nis M c(Y ) +(2.2) +([Voe00, 4.1.7]). +vi. (Duality) Assume that k admits resolution of singularities or the characteristic is +invertible in the coefficients. If T is a smooth scheme of dimension d over k, then +there are canonical isomorphisms +Hom(M(X × T )[n], M c(Y )) ∼= Hom(M(X)(d)[2d + n], M c(Y × T ) +(2.3) +for all n ∈ Z ([FV00, 8.2]). +vii. (Duality) Assume that k admits resolution of singularities or the characteristic is +invertible in the coefficients. +Let Y → X be a flat equidimensional morphism of +relative dimension n of finite type over k. Then there is a canonical morphism of the +form +f ∗ : M c(X)(n)[2n] → M c(Y ). +(2.4) +(Projective bundle formula) If π : E → X is a rank n vector bundle, then (2.4) is an +isomorphism ([Nie08, 2.6]). +6 + +Logarithmic motives with compact support +Moreover, [Voe00, 4.1.8] shows that +M c(An) ≃ Z(n)[2n]. +By (2.2), we get for every scheme X an isomorphism +M c(X × A1) ≃ M c(X)(1)[2] ̸≃ M c(X). +This shows that the motive with compact support is not homotopy invariant. One of our +goals is to create an analogue of the theory of motives with compact support for log schemes +which is homotopy invariant, that is, if we let □ := (P1, ∞) parametrize our homotopies, the +analogue of the motive with compact support for log schemes should satisfy the equivalence +M c(X × □) ≃ M c(X) +for all fs log schemes X. We will show this in corollary 4.13. +2.2. +Logarithmic motivic homotopy theory – Throughout this article, we work in the +framework provided by [BPØ22b], which is the triangulated category of logarithmic motives +over a field k with a trivial log structure. In this section, we give a brief construction and +state basic properties that will be relevant for this article. We will assume familiarity with +the basic properties of logarithmic schemes, for which the standard reference is [Ogu18]. +Let lSm/k denote the category of fine and saturated (abbreviated “fs”) log schemes that +are log smooth over a field k with trivial log structure, and SmlSm/k the full subcategory +consisting of fs log schemes X such that the underlying scheme X is smooth over k. For +X, Y ∈ lSm/k, an elementary log correspondence Z from X to Y consists of +• an integral closed subscheme Z ⊂ X ×Y that is finite and surjective over a component +of X. +• a morphism ZN → Y of fs log schemes, where ZN denotes the fs log scheme whose +underlying scheme is the normalization of Z and the log structure MZN is given as +the pullback p∗ +logMX, where p : ZN → X denotes the induced scheme morphism. +The category lCork of finite logarithmic correspondences over k is the category whose objects +are fs log schemes that are log smooth over k and morphisms between two objects X and Y is +the free abelian group generated lCork(X, Y ) generated by elementary log correspondences +from X to Y . We let +Λltr(X)(−) := lCork(−, X) ⊗ Λ +denote the presheaf with transfers represented by X. +We consider two topologies on lSm/k: First, an analogue of the Nisnevich topology +for log schemes called the strict Nisnevich topology. That is the Grothendieck topology +associated to the cd-structure on lSm/k given by +Y ′ +Y +X′ +X, +g′ +f ′ +f +g +7 + +Nikolai Opdan +where g is an open immersion, f is strict étale (a strict morphism of fs log schemes and an +étale morphism of the underlying schemes), and f −1(X−g(X′)) → X−X′ is an isomorphism +when both sides are considered with the reduced scheme structure. Our prime examples for +strict Nisnevich squares are the left and outer squares of the commutative diagram +Gm +AN +A1 +A1 +□ +P1, +where AN := (A1, 0) and □ := (P1, ∞). +Secondly, we consider the dividing topology: This is the Grothendieck topology asso- +ciated to the cd-structure on lSm/k given by proper log étale surjective monomorphisms +(or equivalently log modifications by ([BPØ22b, A.11.9])). The reason for considering this +topology is that one wants to allow for blow-ups in the log structure without changing +the topological structure. We call the union of these two topologies the dividing Nisnevich +topology. +Letting +□ := (P1, ∞) +be out replacement for A1, one defines the derived category of effective logarithmic motives +logDMeff(k, Λ) +as the homotopy category of C(Shvltr +dNis(k, Λ)) with respect to the □-local descent model +structure ([BPØ22b, B.1.8]). The logarithmic motive M(X) is the image of the sheafification +for the dividing Nisnevich topology of Λltr(X) in logDMeff(k, Λ). +By definition we have monoidal equivalences M(X × Y ) ≃ M(X) ⊗ M(Y ), and espe- +cially □-homotopy invariance M(X × □) ≃ M(X). There is a Mayer-Vietoris property +for strict Nisnevich squares, and every log modification f : Y → X induces an equivalence +of log motives M(f) : M(Y ) ≃ M(X). It is shown in [BPØ22b] that there are generaliz- +ation to log schemes of the classical Gysin triangles ([BPØ22b, 7.5.4]), Blow-up triangles +([BPØ22b, 7.3.3]), the Projective bundle theorem ([BPØ22b, 8.3.5]), and the Thom iso- +morphism ([BPØ22b, 8.3.7]). +Removing the log structure, there is a functor +ω : lSm/k → Sm/k +that sends an object X to X −∂X, where ∂X denotes the set of points on X with nontrivial +log structure. Assuming that k admits resolution of singularities, this functor induces an +adjoint functor pair +ω♯ : logDMeff(k, Λ) +DMeff(k, Λ) : Rω∗, +and Rω∗ is shown to be fully faithful in [BPØ22b, 8.2.13]. Identifying the essential image +of Rω∗ in logDMeff(k, Λ) one finds the smallest subcategory that is closed under small +sums and shifts and contains every motive M(X) where X ∈ lSm/k is proper over k. How- +ever, since the sheaf of logarithmic differentials Ωj +/k are representable in logDMeff(k, Λ) +([BPØ22b, 9.7.1]), but not in DMeff(k, Λ), this shows that Rω∗ is not essentially surjective. +8 + +Logarithmic motives with compact support +We briefly mention the paper [BM21] which shows that logDMeff(k, Λ) admits a t- +structure which induces the t-structure on DMeff(k, Λ) under ω♯. They generalize the clas- +sical purity theorem and Morel’s connectivity theorem to the logarithmic setting. However, +we will not make use of any of these results. +3. +Logarithmic correspondences with compact support +3.1. +Finite logarithmic correspondences with compact support – To introduce the +logarithmic motive with compact support we follow the classical approach by first introduce +logarithmic correspondences with compact support. +Definition 3.1. For X ∈ lSm/k and Y ∈ lSch/k, an elementary logarithmic correspondence +with compact support Z from X to Y is +i. an integral closed subscheme Z ⊂ X×Y such that the projection Z → X is quasi-finite +and dominant. +ii. a morphism ZN → Y of fs log schemes, where ZN denotes the fs log scheme whose +underlying scheme is the normalization of Z, and if p : ZN → X denotes the induced +scheme morphism, then the log structure MZN is given as p∗ MX. +For any ring Λ, the group lCorc +k(X, Y ) ⊗Z Λ is the free abelian Λ-module generated by +elementary logarithmic correspondence with compact support between X and Y . +Remark 3.2. This definition is a rephrasing of the definition of correspondences with com- +pact support given in [SV00] (see also [Blo86]) adapted to logarithmic geometry. The first +condition is a direct translation of the standard definition, while the second condition is +necessary to construct compositions of log correspondences with compact support. +We +note that the second condition is equivalent to the second condition in Definition 2.1.1 in +[BPØ22b]. +Some properties are immediate: +i. Since the underlying scheme Z of a log correspondence Z is finite and surjective over +a component of X, we have +lCork(X, Y ) ⊂ lCorc +k(X, Y ). +ii. If Y has trivial log structure, then the second condition is trivially satisfied, and +lCorc +k(X, Y ) ≃ zequi(Y, 0)(X), +where zequi(−, r) is sheaf of relative dimension r cycles as defined in [SV00]. +iii. If X has trivial log structure, then ZN has trivial log structure, and ZN → Y factors +through Zn → Y − ∂Y . Hence +lCorc +k(X, Y ) ≃ Corc +k(X, Y − ∂Y ). +iv. If both X and Y has trivial log structure, then combining the two above properties +we get +lCorc +k(X, Y ) ≃ Corc +k(X, Y ). +9 + +Nikolai Opdan +Definition 3.3. For X ∈ lSch/k we define a presheaf of Λ-modules with transfers Λc +ltr(X) +on lCork by +Y �→ Λc +ltr(X)(Y ) := lCorc +k(Y, X) ⊗ Λ. +Given a proper morphism f : S′ → S, the pullback of cycles induces a natural map +f ∗ : Λc +ltr(X)(S) → Λc +ltr(X)(S′). +On the other hand, a flat morphism g : S′ → S of relative dimension 0 (an open immersion) +induces a map +g∗ : Λc +ltr(X)(S) → Λc +ltr(X)(S′). +The injection Λltr(X)(−) ֒→ Λc +ltr(X)(−) makes the representable sheaf Λltr(X) a sub- +sheaf of Λc +ltr(X) for all X since the structure morphism associated to U → V is compatible +with log transfers. +Remark 3.4. If p : ZN → X and q : ZN → Y denotes the underlying scheme morphisms, +then referring to Definition III.1.1.5 in [Ogu18], the second condition of definition 3.1 is +equivalent to the giving a morphism +q∗ +log MY → p∗ +log MX +[BPØ22b, 2.1.2]. +From this, we see that Λc +ltr(ptN) is the empty sheaf since there can be no such morphism +q. +Proposition 3.5. Let X be an fs log scheme. If X is proper, then Λc +ltr(X) = Λltr(X). +Proof. If X is proper, the underlying scheme X is proper as well. This implies that the +underlying scheme Z of a finite log correspondence Z ∈ Λc +ltr(X)(U) is proper over U. Since +Z now is both quasi-finite and proper, [Sta22, 30.21.1] shows that it is also finite over U. +We conclude that Z ∈ Λltr(X)(U). +Definition 3.6. A coherent log scheme X over k is said to be solid if for any point x ∈ X +the induced map +Spec (OX,x) → Spec (MX,x) +is surjective. +Lemma 3.7. Let f : X → Y be a quasi-finite and dominant morphism of fs log schemes. +If Y is smooth over k, then X is solid. +Proof. Owing to Lemma 2.2.2 and Lemma 2.2.7 in [BPØ22b] it suffices to show that f is +an open immersion. Since Y is normal by Lemma 2.2.8 in [BPØ22b], we conclude that f is +universally open by Lemma 37.67.2 in [Sta22]. +Lemma 3.8. Let X, Y ∈ lSm/k and suppose that Z is a closed subscheme of X × Y . Then +there exists at most one elementary log correspondence with compact support whose under- +lying scheme is Z. +Proof. Using lemma 3.7, this follows the proof of Lemma 2.3.1 in [BPØ22b]. +10 + +Logarithmic motives with compact support +Lemma 3.9. Let X and Y be fs log schemes that are log smooth over k. Then the homo- +morphism of abelian groups +lCorc +k(X, Y ) −→ lCorc +k(X − ∂X, Y − ∂Y ), +defined by V �→ V − ∂V , is injective. +Proof. Using lemma 3.8 this follows the proof of Lemma 2.3.2 in [BPØ22b]. +Proposition 3.10. Let X be an fs log scheme log smooth over k. Then Λc +ltr(X) is a strict +étale sheaf. +Proof. This is essentially the proof of Proposition 4.5.1 in [BPØ22b]. +Proposition 3.11. For every X ∈ lSm/k, the sheaf a∗ +dNisΛc +ltr(X) is a dividing Nisnevish +sheaf with log transfers. +Proof. Since Λc +ltr(X) is a strict étale sheaf by proposition 3.10, we may apply the dividing +Nisnevich sheafication functor +a∗ +dNis : Pshlog(Sm/k) −→ Shvlog(Sm/k). +The result now follows from [BPØ22b, 4.5.7] which states that the dividing Nisnevich topo- +logy is compatible with log transfers. +Definition 3.12. [BPØ22b, Definition 7.6.1] An admissible blow-up is a proper birational +map +X′ → X +of fs log schemes over k such that the induces morphism +X′ − ∂X′ → X − ∂X +is an isomorphism. +We let ABl /k denote the class of admissible blow ups, and we let +(ABl /k) � Y denote the class of all admissible blow-ups over Y . +Proposition 3.13. Assume that k admits resolution of singularities. Let X be a smooth +scheme over k and Y an fs log scheme log smooth over k. Then there is a naturally induced +isomorphism +colim +Y ′∈(ABl /k)�Y Λc +ltr(X)(Y ′) ≃ Λc +ltr(X)(Y − ∂Y ). +Proof. Following the proof of Proposition 8.2.1 [BPØ22b], we let Y ′ → Y be an admissible +blow-up. The induced isomorphism Y ′ − ∂Y ′ → Y − ∂Y allows for the construction of a the +composite homomorphism +ϕY ′ : Λc +ltr(X)(Y ′) → Λc +ltr(X)(Y ′ − ∂Y ) ≃ Λc +ltr(X)(Y − ∂Y ). +Gathering the ϕY ′ we get a morphism +ϕ : +colim +Y ′∈(ABl /k)�Y Λc +ltr(X)(Y ′) → Λc +ltr(X)(Y − ∂Y ). +Let W ∈ Λc +ltr(X, Y ′) be an elementary log correspondence with compact support. If +ϕY ′(W) = 0, then W = 0, since W is the closure of ϕY ′(W) ∈ Y ′ × X. +11 + +Nikolai Opdan +For the surjectivity, let Z ∈ Λc +ltr(X, Y − ∂Y ). By Raynaud–Gruson platification ([RG71, +5.2.2]), there is a scheme Y ′ and a morphism f : Y ′ → Y such that f is an isomorphism +on Y − ∂Y and the closure of Z′ of Z ⊂ Y ′ × X is flat over Y ′. The subscheme Z′ is a +correspondence with compact support from Y ′ to X. +By resolution of singularities there is a blow up g : Y ′′ → Y ′ such that Y ′′ is smooth +over k, and the complement of (f ◦ g)−1(Y − ∂Y ) in Y ′′ consists of strict normal crossing +divisors Z1, . . . , Zr. Letting Y ′′ = (Y ′′, Z1 + . . . + Zr), the induced morphism Y ′′ → Y of +fs log schemes is an admissible blow-up, and the closure Z′′ of Z in Y ′′ × X is a closed +subscheme of Z′ ×Y ′ Y ′′, which preserved under base change, is quasi-finite. It follows that +Z can be extended to a finite correspondence with compact support from Y ′′ to X. Since +X has trivial log structure, this gives a log correspondence with compact support from Y ′′ +to X. Thus ϕ is surjective. +Lemma 3.14. If f ′ : X′ → X is a log modification in lSm/k. For every Y ∈ lSm/k and +V ∈ lCorc +k(Y, X) there exists a dividing Zariski cover g′ : Y ′ → Y and W ∈ lCorc +k(Y ′, X′) +fitting in a commutative diagram +Y ′ +X′ +Y +X. +W +g +f +V +Proof. Similar to the proof for Lemma 4.5.5 in [BPØ22b]. +Recall from [BPØ22b] the adjuction of categories +Pshlog(k, λ) +Pshltr(k, λ) +Shvlog(k, λ) +Shvltr(k, λ). +γ♯ +a∗ +dNis +γ∗ +a∗ +dNis +γ♯ +adNis ∗ +γ∗ +adNis ∗ +Proposition 3.15. Suppose that f : Y → X is an log modification of fs log schemes. Then +the induced morphism +a∗ +dNisγ∗Λc +ltr(Y ) −→ a∗ +dNisγ∗Λc +ltr(X) +of dividing Nisnevich sheaves is an isomorphism. +Proof. For any fs log scheme T there is a commutative diagram +Λc +ltr(Y )(T ) +Λc +ltr(X)(T ) +Λc +tr(Y − ∂Y )(T − ∂T ) +Λc +tr(X − ∂X)(T − ∂T ). +The vertical morphisms are injections by lemma 3.9, and since the lower morphism is an +isomorphism since Y − ∂Y ≃ X − ∂X, the top morphism is injective as well. +Hence +Λc +ltr(Y ) → Λc +ltr(X) is a monomorphism. This implies that +a∗ +dNisγ∗Λc +ltr(Y ) −→ a∗ +dNisγ∗Λc +ltr(X) +12 + +Logarithmic motives with compact support +is a monomorphism since a∗ +dNis and γ∗ are exact by [BPØ22b, 4.2.10]. +Due to lemma 3.14 there exists a dividing Nisnevich cover g : T ′ → T and a finite log +correspondence with compact support W ∈ Λc +ltr(Y )(T ′) such that f ◦ W = g ◦ V for every +V ∈ Λc +ltr(Y )(T ). This proves that a∗ +dNisΛc +ltr(Y ) −→ a∗ +dNisΛc +ltr(X) is an epimorphism, which +finishes the proof. +4. +Logarithmic motives with compact support +4.1. +Definition of logarithmic motives with compact support – In this section, we +introduce the logarithmic motive with compact support and establish basic properties using +the results above. +Definition 4.1. For any X ∈ lSm/k, the logarithmic motive with compact support of X, +denoted M c(X), is the image of a∗ +dNisΛc +ltr(X) in logDMeff(k, Λ). Given a morphism of fs +log schemes f : Y → X, we let +M c(Y +f→ X) +denote the cone in logDMeff(k, Λ) associated with the morphism of complexes +a∗ +dNisΛc +ltr(Y ) → a∗ +dNisΛc +ltr(X) +in C(Shvltr +dNis(k, Λ)). We also denote it by M c(Y → X) if the morphism is understood, or +sometimes simply M c(f). +The inclusion Λltr(X) ֒→ Λc +ltr(X) induces a canonical map M(X) ֒→ M c(X). If X is +proper, proposition 3.5 implies that this map is an isomorphism, i.e. we have +M c(X) ≃ M(X). +(4.1) +Proposition 4.2. Suppose that +Y ′ +Y +X′ +Y +is a dividing Nisnevich distinguished square. Then there is an induced isomorphism +M c(Y → X) +≃ +−→ M c(Y ′ → X′). +Proof. We need to show that for every strict □-invariant complex F of dividing Nisnevich +sheaves with log transfers there is an isomorphism +HomlogDMeff(k,Λ)(M c(Y → X), F) −→ HomlogDAeff(k,Λ)(M c(Y ′ → X′), F). +We may further assume that F is a fibrant object in C(Shvltr +dNis(k, Λ)) with regard to the +□-local descent structure. By arguing similar to [BPØ22b, 5.2.3] this reduces the problem +to showing that there is an isomorphism +HomD(Shvltr +dNis(k,Λ))(a∗ +dNisΛc +ltr(Y → X), F) ≃ HomD(Shvltr +dNis(k,Λ))(a∗ +dNisΛc +ltr(Y ′ → X′), F). +13 + +Nikolai Opdan +The result now follows if we can show that the exactness of the sequence of dividing +Nisnevich sheaves with log transfers +0 → a∗ +dNisΛc +ltr(Y ) +f→ a∗ +dNisΛc +ltr(Y ′) ⊕ a∗ +dNisΛc +ltr(X) +g→ a∗ +dNisΛc +ltr(X′) → 0. +(4.2) +For [BPØ22b, 4.3.5] the sequence +0 → a∗ +dNisΛltr(Y ) → a∗ +dNisΛltr(Y ′) ⊕ a∗ +dNisΛltr(X)→a∗ +dNisΛltr(X′) → 0 +is exact, and using the canonical inclusions Λltr(X) → Λc +ltr(X) we get a commutative dia- +gram +0 +a∗ +dNisΛc +ltr(Y ) +a∗ +dNisΛc +ltr(Y ′) ⊕ a∗ +dNisΛc +ltr(X) +a∗ +dNisΛc +ltr(X′) +0 +0 +a∗ +dNisΛltr(Y ) +a∗ +dNisΛltr(Y ′) ⊕ a∗ +dNisΛltr(X) +a∗ +dNisΛltr(X′) +0, +f +g +which proves the injectivity of f. The exactness at a∗ +dNisΛc +ltr(Y ′)⊕a∗ +dNisΛc +ltr(X) follows from +the commutativity of the diagram. For the surjectivity of g we choose compactifications +X′ → X′, Y ′ → Y ′, X → X, and y → Y +and get a commutative diagram +0 +a∗ +dNisΛltr(Y ) +a∗ +dNisΛc +ltr(Y ′) ⊕ a∗ +dNisΛc +ltr(X) +a∗ +dNisΛc +ltr(X) +0 +0 +a∗ +dNisΛc +ltr(Y ) +a∗ +dNisΛc +ltr(Y ′) ⊕ a∗ +dNisΛc +ltr(X) +a∗ +dNisΛc +ltr(X′) +0. +f +g +The surjectivity of g follows from the vertical morphisms’ surjectivity. +This proves the +exactness of (4.2) which completes the proof. +4.2. +Gysin triangles – In [Voe00, 3.5.4] Voevodsky constructed a Gysin triangle for mixed +motives over a perfect field k. That is, for a closed isomorphism i : Z → X between smooth +k-schemes, there is a distinguished triangle +M(X − Z) → M(X) +i∗ +→ M(Z)(n)[2n] → M(X − Z)[1]. +This triangle corresponds to the localization sequence in cohomology, which is an important +tool in studying higher Chow groups. This section aims to prove a weak analog for log +schemes that generalize parts of the classical result. Namely, we will prove the following: +Theorem 4.3. Let X be a smooth scheme over k and suppose that Z1, Z2, · · · , Zr are smooth +irreducible divisors forming a strict normal crossing divisor Z1 + . . . + Zr on X ∈ Sm/k. +Let Zs,t := Zs + Zs+1 + . . . + Zt. Then there exists a distinguished triangle +M c(X, Z1,r) +M c(X, Z1,r−1) +M c(Zr, Zr ∩ Z1,r−1)(1)[2] +. . . ++1 +in logDMeff(k, Λ). +14 + +Logarithmic motives with compact support +Proof. Suppose that Z is a smooth irreducible divisor on X. Applying the technique of +deformation to the normal cone in [BPØ22b, 7.2.8], originally due to Fulton [Ful84], we +Nisnevich locally on X have a Cartesian diagram +Z +X +As +As+1, +u +where u is an étale morphism and s denotes the dimension of Z over k. Let X1 = As +Z and +∆ be the diagonal of Z over As, and let +X2 := X ×As X1 − (X ×As +1 Z − ∆) ∪ (Z ×As+1 X1 − ∆). +Let DZX denote the deformation space (Definition 7.4.1 in [BPØ22b]) of the pair (X, Z). +Arguing similar to [BPØ22b, 7.5.4], since blow-ups commute with flat base change, we +have a Cartesian square +DZX2 +X2 × □ +DZX +X × □ . +This implies that the map +DZX2 → DZX +is étale. Furthermore, since X2 → X and NZX2 → NZX are also étale, where NZX is the +normal bundle of Z in X. Strict Nisnevich descent then implies that there are isomorphisms +M c((BlZX) → X) +≃ +−→ M c((BlZX2, E2) → X2) +M c((BlZ×□(DZX)) → DZX) +≃ +−→ M c((BlZ×□(DZX2), ED +2 ) → DZX2) +M c((BlZ(NZX)) → NZX) +≃ +−→ M c((BlZ(NZX2), EN +2 ) → NZX2). +Applying deformation to the normal cone, we can replace X by X2, and similarly, we can +repeat the argument to replace X2 by X1. This reduces this case to proving it for X = A1 +and Z = 0. +Using the square (2.2) and strict Nisnevich descent (proposition 4.2) we have isomorph- +isms +M c((Bl0P1, E) → P1) +≃ +−→ M c((Bl0 A1, E) → A1). +Invoking Theorem 7.5.4 in [BPØ22b] we get the distinguished triangle +M c(Bl0P1, E) → M c(P1) → Λ(1)[2] → M c(Bl0P1, E), +hence the distinguished triangle +M c(Bl0 A1, E) → M c(A1) → Λ(1)[2] → M c(Bl0 A1, E). +Applying dividing descent (proposition 4.2) to the log modification +(Bl0 A1, E) → (A1, 0) +15 + +Nikolai Opdan +we have the desired triangle +M c(A1, 0) → M c(A1) → M c(0)(1)[2] → M c(A1, 0). +If Z = Z1 + . . . + Zr with each Zi is irreducible, we by induction we a diagram +M c(X, Z1,r) +M c(X, Z2,r) +M c(Z1, Z1 ∩ Z2,r)(1)[2] +. . . +M c(X, Z1,r−1) +M c(X, Z2,r−1) +M c(Z1, Z1 ∩ Z2,r−1)(1)[2] +. . . +M c(Zr, Zr ∩ Z1,r−1)(1)[2] +M c(Zr, Zr ∩ Z2,r−1)(1)[2] +M c(Z1 ∩ Zr)(2)[4] +. . . +... +... +... ++1 ++1 ++1 ++1 ++1 ++1 +where each square is commutative, and where the two bottom rows and two right columns +are distinguished triangles. Then the 3×3-lemma ([May01, 2.6]) assures that the top and +left triangles are cofibre sequences as well. +Corollary 4.4. Let Z be a smooth irreducible divisor on a smooth scheme X. Then there +exists a distinguished triangle +M c(X, Z) → M c(X) → M c(Z)(1)[2] → M c(X, Z)[1] +in logDMeff(k, Λ). +Remark 4.5. Consider the open immersion A1 ֒→ □. One cannot yet state a localization +sequence analogous to the classical case (2.1). After all, it would involve the definition of +the motive of a point with a nontrivial log structure, which is not defined because it is not +log smooth. In lack of an analogy for this important theorem, we will often instead be able +to use the above corollary and the general theorem. +4.3. +Künneth formula – Classically, the Künneth formula [Voe00, 4.1.7] for motives with +compact support is a direct consequence of the localization theorem [Voe00, 4.1.5]. However, +we prove it directly since we do not yet have such a result (see remark 4.5). +Let +v : lCork → lCork[(ABl /k)−1] +denote the localization functor where ABl /k denotes the class of admissible blow-ups that +were defined in definition 3.12. Then there exists functors +C≤0(Psh(lCork, Λ)) +C≤0(Psh(lCork[(ABl /k)−1], Λ)) +v♯ +v∗ +v∗ +where v♯ is left adjoint to v∗ and v∗ is left adjoint to v∗. +Proposition 4.6. Assume that k admits resolution of singularities. +For every smooth +scheme X over k, there is an isomorphism +a∗ +dNisΛc +ltr(X) ≃ ω∗Λc +tr(X). +16 + +Logarithmic motives with compact support +Proof. By Lemma 4.4.3 in [BPØ22b] there is an isomorphism +a∗ +dNisv∗v♯Λc +ltr(X) ≃ v∗v♯a∗ +dNisΛc +tr(X). +Applying proposition 3.13 and using the identification (6.2.1) in [BPØ22b] we get an iso- +morphism +a∗ +dNisv∗v♯Λc +ltr(X) ≃ ω∗a∗ +dNisΛc +ltr(X), +and using Remark 7.6.8 in [BPØ22b] we have +v∗v♯a∗ +dNisΛc +ltr(X) ≃ a∗ +dNisΛc +ltr(X). +We conclude by the face that a∗ +dNisΛc +ltr(X) is a dividing Nisnevich sheaf, since the functor +η♯ : Shv(két, Λ) → Shvl ét(k, Λ) +is fully faithful (Lemma 8.5.2 in [BPØ22b]), where η : két → lSm/k is the inclusion functor. +Definition 4.7. Recall from [BPØ22b, 156] the definition of the complex C∗ F for a +presheaves with log transfers F and U ∈ lSm/k as +C∗ F(U) := +� +· · · → +colim +(Y ∈ABl)�(U×∆ 1 +□ ) F(Y ) → +colim +(Y ∈ABl)�U F(Y ) +� +Similarly, if F is bounded above we define the double Suslin complex for U ∈ Sm/k as +CA1 +∗ F(U) := +� +· · · → F(U × ∆1 +A1) → F(U) +� +where +∆n +A1 := Spec k[x0, . . . , xn]/ +� n +� +i=0 +xi = 1 +� +is the standard A1-simplex. +Lemma 4.8. Assume that k admits resolution of singularities and let X be a smooth scheme +over k. Then there are isomorphisms +ω∗Λc +tr(X) ≃ C∗ω∗Λc +XCA1 +∗ ω∗Λc +X. +Proof. The first isomorphism comes from Proposition 6.2.4(4) and Remark 7.6.8 in +[BPØ22b]. The second isomorphism comes from applying [BPØ22b, 8.2.3]. +The following result is a generalization of [BPØ22b, 8.2.6] to non-proper schemes X. +Theorem 4.9. Assume that k admits resolution of singularities. Let X ∈ Sm/k and Y ∈ +lSm/k. Then for any n ∈ Z there is an isomorphism +HomlogDMeff(k,Λ)(M(Y )[n], M c(X)) ≃ HomDMeff(k,Λ)(M(Y − ∂Y )[n], M c(X)). +17 + +Nikolai Opdan +Proof. Lemma 4.8 gives an equivalence between +HomlogDMeff(k,Λ)(M(Y )[n]), ω∗Λc +tr(X)) +and +HomlogDMeff(k,Λ)(M(Y )[n]), ω∗CA1 +∗ Λc +tr(X)). +The complex CA1 +∗ Λc +tr(X) is strictly A1-invariant, hence it is strictly □-invariant in the divid- +ing Nisnevich topology. Applying Proposition 5.2.3 in [BPØ22b] there is an isomorphism +HomlogDMeff(k,Λ)(M(Y ), ω∗CA1 +∗ Λc +tr(X)[n]) ≃ Hn +dNis(Y, ω∗CA1 +∗ Λc +tr(X)). +Further applying Lemma 8.1.10 in [BPØ22b] we have +Hn +dNis(Y, ω∗CA1 +∗ Λc +tr(X)) ≃ Hn +Nis(Y − ∂Y, CA1 +∗ Λc +tr(X)), +which by Proposition 14.16 gives an isomorphism +Hn +Nis(Y − ∂Y, CA1 +∗ Λc +tr(X)) ≃ HomDMeff(k,Λ)(M(Y − ∂Y ), M c(X)[n]). +Moving the shift gives the right isomorphism. +Corollary 4.10. Assume that k admits resolution of singularities, and let X be a smooth +scheme over k. Then the unit of adjunction id → Rω∗ω♯ induces an equivalence +M c(X) ≃ Rω∗ω♯M c(X). +Proof. It suffices to show that for every generator M(Y )[i] for Y ∈ lSm/k and i ∈ Z that +there is an isomorphism +Hom(M(Y )[i], M(X)) ≃ Hom(M(Y )[i], Rω∗M(X)). +Using the isomorphisms +Hom(M(Y )[i], Rω∗M(X)) ≃ HomDMeff(k,Λ)(ω♯M(Y )[i], M(X)) +≃ HomDMeff(k,Λ)(M(Y − ∂Y )[i], M(X)), +the result follows from theorem 4.9. +Example 4.11. Taking X = A1 in the corollary we get the isomorphisms +M c(A1) ≃ Rω∗ω♯M c(A1) ≃ Rω∗M c(A1) ≃ Rω∗Λ(1)[2]. +This gives the calculation +M c(A1) ≃ Λ(1)[2] +(4.3) +in logDMeff(k, Λ). +Theorem 4.12. Assume that k admits resolution of singularities. +Let X and Y be log +smooth fs log schemes over k. Then there is an canonical equivalence +M c(X) ⊗ M c(Y ) +≃ +−→ M c(X × Y ) +in logDMeff(k, Λ). +18 + +Logarithmic motives with compact support +Proof. Step 1: Assuming resolution of singularities, and for X, Y ∈ Sm/k, we have +ω∗(M c(X)) ⊗ ω∗M c(Y )) ≃ ω∗M c(X × Y ) +by the classical Künneth formula ([Voe00, 4.1.7]). The result then follows by applying Rω∗ +and using Corollary 4.10. If X and Y are proper fs log schemes it follows from the monoidal +structure in logDMeff(k, Λ) using that M c(X) = M(X). +Step 2: Assume that X ∈ SmlSm/k and Y ∈ Sm/k. By [BPØ22b, A.5.10] we can +assume that X is of the form (X, Z), where Z is a strict normal crossing divisor on X. +Writing Z = Z1 + . . . Zr we use induction on r to find a diagram of cofibre sequences +M c(X, Z1 + . . . + Zr) ⊗ M c(Y ) +M c(X × Y, (Z1 + . . . + Zr) × Y ) +M c(X, Z1 + . . . + Zr−1) ⊗ M c(Y ) +M c(X × Y, (Z1 + . . . + Zr−1) × Y ) +M c(Zr, Zr ∩ (Z1 + . . . + Zr−1))(1)[2] ⊗ M c(Y ) +M c(Zr × Y, (Zr ∩ (Z1 + . . . + Zr)) × Y )(1)[2]. +The induction hypothesis shows that the two lower horizontal maps are isomorphisms; hence +we conclude that the top morphism is also an isomorphism. +Step 3: Assume that X ∈ lSm/k and Y ∈ Sm/k. Then by toric deformation ([BPØ22b, +A.10.2]) and proposition 4.2 we reduce to Step 2. +By symmetry, we can apply the arguments to Y as well when X ∈ Sm/k. +Step 4: It remains to show the general case, i.e., for X, Y ∈ lSm/k. +Writing Z = +Z1 + . . . + Zr and W = W1 + . . . + Ws we find that applying theorem 4.3 twice gives us a +commutative diagram +M c(X, Z1,r) ⊗ M c(Y, W1,s) +M c(X × Y, Z1,r × Y + W1,s × X) +M c(X, Z1,r−1) ⊗ M c(Y, W1,s−1) +M c(X × Y, Z1,r−1 × Y + W1,s−1 × X) +M c(Zr, Zr ∩ Z1,r−1)(1)[2] ⊗ M c(Ws, Ws ∩ W1,s−1)(1)[2] +M c(Zr × Ws, (Zr ∩ Z1,r−1) × Ws + (Ws ∩ W1,s−1) × Zr)(2)[4]. +By double induction on r and s, we find that the two lower horizontal morphisms are +isomorphisms, proving that the top horizontal morphism is also an isomorphism. +Step 2 in the proof is essential as it proves homotopy invariance of the log motive with +compact support. This marks a significant differentiation from the classical theory, which +is not homotopy invariant (A1-invariant). +Using calculations in [BPØ22b], we make some easy computations in the following corol- +lary. +Corollary 4.13. Assume that k admits resolution of singularities. Then for any X ∈ lSm/k +there are isomorphisms +M c(X × □) ∼= M c(X), +(4.4) +M c(X × (Pn+1, Pn)) ∼= M c(X), +(4.5) +and +M c(X × Pn) ∼= M c(X) ⊗ +n +� +i=0 +Λ(i)[2i]. +(4.6) +19 + +Nikolai Opdan +Proof. The first and second isomorphisms uses □-invariance and (Pn+1, Pn) of the log motive. +The final isomorphism follows from the calculation +M(Pn) ≃ +n +� +i=0 +Λ(i)[2i] +from Proposition 8.2.6 in [BPØ22b]. +4.4. +Duality – One uses the motive with compact support to state a duality theorem for +non-proper schemes. We will establish the same statement for fs log schemes. +Remark 4.14. Although the motive of A1 is unknown, it follows from work of Saito that +the category of reciprocity sheaves embeds fully faithfully in logDMeff(k, Λ) ([Sai21]), that +the relations between the Witt-vectors and M(A1) forces M(A1) to be very big. Using that +M c(A1) is merely Z(1)[2], this gives an affirmative answer to the question raised in [BPØ22b, +8.2.7] that the equivalence +HomlogDMeff(k,Λ)(M(Y )[i], M(X)) ≃ HomDMeff(k,Λ)(M(Y − ∂Y )[i], M(X)) +does not necessarily hold for non-proper schemes X. +One then has the analogue of the duality theorem [FV00, 8.2] for fs log schemes: +Theorem 4.15. Assume that k admits resolution of singularities. +Let X ∈ Sm/k and +Y, T ∈ lSm/k such that T is of pure dimension d over k. Then for any n ∈ Z there is an +isomorphism +Hom(M(Y × T )[n]), M c(X)) ≃ Hom(M(Y )(d)[2d + n]), M c(X × (T − ∂T ))) +in logDMeff(k, Λ). +Proof. By theorem 4.9 there is a isomorphism between +HomlogDMeff(k,Λ)(M(Y × T )[n], M c(X)) +and +HomDMeff(k,Λ)(M((Y − ∂Y ) × (T − ∂T ))[n], M c(X)). +Using the classical duality theorem ([FV00, 8.2]) by Friedlander and Voevodsky, there is +further an isomorphism to +HomDMeff(k,Λ)(M(Y − ∂Y )(d)[2d + n], M c(X × (T − ∂T ))). +Reapplying theorem 4.9 we get an isomorphism to +HomlogDMeff(k,Λ)(M(Y )(d)[2d + n], M c(X × (T − ∂T ))). +The composition of these isomorphisms give the desired isomorphism. +20 + +Logarithmic motives with compact support +4.5. +Cancellation – Cancellation is classically known as the equivalence +HomDMeff(k,Λ)(M, N) ≃ HomDMeff(k,Λ)(M(1), N(1)) +induced by tensoring with Λ(1), also known as Gm-cancellation. The fact that this is true for +perfect fields k is a difficult theorem due to Voevodsky in [Voe10]. However, if one assumes +resolution of singularities, a much simpler proof is available using the motive with compact +support ([Voe00, 4.3.1]). +Assuming resolution of singularities, the cancellation property in the full subcategory +logDMeff +prop(k, Λ) of logDMeff(k, Λ) generated by fs log schemes whose underlying scheme +is proper over k follows the fact that the +Rω∗ : DMeff(k, Λ) −→ logDMeff(k, Λ) +is fully faithful by [BPØ22b, 8.2.13] and from the identification of its essential image as +logDMeff +prop(k, Λ) by [BPØ22b, 8.2.17]. We will instead combine almost all our above results +to give an alternative easier proof without relying on these results. +Theorem 4.16. Assume that k admits resolution of singularities. Let M and N be two +objects of logDMeff +prop(k, Λ). Then tensoring with Λ(1) induces an isomorphism +Hom(M, N) ≃ Hom(M(1), N(1)). +Proof. We reduce to the case when M, N ∈ SmlSm/k and begins by assuming that N = +M(X, Z) where X is a smooth scheme over k and Z is a smooth irreducible divisor on X. +Then corollary 4.4 gives two cofiber sequences sitting in a commutative square +Hom(M, M(X, Z)) +Hom(M(1), M(X, Z)(1)) +Hom(M, M(X)) +Hom(M(1), M(X)(1)) +Hom(M, M(Z)(1)[2]) +Hom(M(1), M(Z)(2)[2]), +where the two lower horizontal injections are the result of the composition of the morphisms +Hom(M(Y ), M(X)) ֒→ Hom(M(Y ), M c(X)) +4.9 +≃ HomDMeff(k,Λ)(M(Y − ∂Y ), M(X)) +≃ HomDMeff(k,Λ)(M(Y − ∂Y )(1), M c(X)(1)). +These maps are all isomorphisms if X is proper. +Applying induction to the number of irreducible components of +Z = Z1 + . . . + Zr, +we use theorem 4.3 twice to get a commutative diagram +21 + +Nikolai Opdan +Hom(M, M(X, Z1 + . . . Zr)) +Hom(M(1), M(X, Z1 + . . . Zr)(1)) +Hom(M, M(X, Z1 + . . . Zr−1)) +Hom(M(1), M(X, Z1 + . . . Zr−1)(1)) +Hom(M, M(Zr, Zr ∩ (Z1 + . . . Zr−1))(1)[2]) +Hom(M(1), M(Zr, Zr ∩ (Z1 + . . . Zr−1))(2)[2])), +The induction hypothesis implies that the two bottom horizontal morphisms are iso- +morphisms, which also shows that the top morphism is an isomorphism. +Question 4.17. Does the cancellation theorem hold in greater generality in logDMeff(k, Λ)? +5. +Logarithmic invariants with compact support +5.1. +Logarithmic Borel–Moore homology and logarithmic motivic cohomology with +compact support – Having defined the logarithmic motive with compact support and +worked out some properties, we derive new invariants for logarithmic schemes. +Definition 5.1. For any fs log scheme X and any ring Λ, we define logarithmic Borel–Moore +homology by +HlBM +n,i +(X, Λ) := HomlogDMeff(k,Λ)(Λ(n)[i], M c(X)), +and logarithmic motivic cohomology with compact support by +Hn,i +lc (X, Λ) := HomlogDMeff(k,Λ)(M c(X), Λ(n)[i]). +Because of the results from the previous sections, we easily deduce some properties. We +state these for logarithmic Borel–Moore homology. +Properties 5.2. Let X, Y, Z be fs log schemes log smooth over k and n, i be positive integers. +i. (Contravariant functoriality) For any flat morphism f : X → Y of relative dimension +0 we have an induced morphism +f ∗ : HlBM +n,i +(Y, Λ) → HlBM +n,i +(X, Λ). +ii. (Covariant functoriality) For any proper map g : Z → Y of relative dimension 0 there +is a map +g∗ : HlBM +n,i +(Z, Λ) → HlBM +n,i +(Y, Λ). +iii. (Homotopy invariance) Assume that k admits resolution of singularities. For any X +there is an isomorphism +HlBM +n,i +(X × □, Λ) ≃ HlBM +n,i +(X, Λ) +by Corollary 4.13. +iv. (Weak Gysin triangle) If V is a smooth irreducible divisor in X we find by applying +Corollary 4.4 that there is a distinguished triangle +HlBM +n,i +((X, Z), Λ) → HlBM +n,i +(X, Λ) → HlBM +n+2,i+1(Z, Λ) → HlBM +n+1,i((X, Z), Λ). +(5.1) +22 + +Logarithmic motives with compact support +v. Assume that k admits resolution of singularities. If W is a smooth scheme and j is a +positive integer we get an isomorphism +HlBM +n,i +(W, Λ) ≃ HlBM +n+j,i+2j(W, Λ). +(5.2) +by Theorem 4.15. +References +[ATW19] +D. Abramovich, M. Temkin and J. Włodarczyk. ‘Functorial embedded resolution +via weighted blowings up’. Preprint (2019). arXiv: 1906.07106. +[Blo86] +S. Bloch. ‘Algebraic cycles and higher K-theory’. Advances in Mathematics 61.3 +(1986), 267–304. +[BM21] +F. Binda and A. Merici. ‘Connectivity and Purity for Logarithmic Motives’. +Journal of the Institute of Mathematics of Jussieu (2021), 1–47. arXiv: 2012.08361. +[BPØ22a] +F. Binda, D. Park and P. A. Østvær. ‘Motives and homotopy theory in logar- +ithmic geometry’. C. R., Math., Acad. Sci. Paris 360 (2022), 717–727. +[BPØ22b] +F. Binda, D. Park and P. A. Østvær. Triangulated categories of logarithmic +motives over a field. Vol. 433. Astérisque. Paris: Société Mathématique de France +(SMF), 2022. arXiv: 2004.12298. +[Ful84] +W. Fulton. Intersection theory. Vol. 2. Ergebnisse der Mathematik und ihrer +Grenzgebiete. 3. Folge. Berlin: Springer, 1984. +[FV00] +E. M. Friedlander and V. Voevodsky. ‘Bivariant cycle cohomology’. Cycles, +transfers, and motivic homology theories. Vol. 143. Ann. of Math. Stud. Prin- +ceton Univ. Press, Princeton, NJ, 2000, 138–187. +[Hir64] +H. Hironaka. ‘Resolution of Singularities of an Algebraic Variety Over a Field +of Characteristic Zero’. Annals of Mathematics 79.1 (1964), 109–203. +[May01] +J. May. ‘The Additivity of Traces in Triangulated Categories’. Advances in +mathematics 163 (2001), 34–73. +[Nie08] +Z. Nie. ‘Karoubi’s Construction for Motivic Cohomology Operations’. American +Journal of Mathematics 130.3 (2008), 713–762. +[Ogu18] +A. Ogus. Lectures on Logarithmic Algebraic Geometry. Cambridge Studies in +Advanced Mathematics. Cambridge University Press, 2018. +[RG71] +M. Raynaud and L. Gruson. ‘Critres de platitude et de projectivit: Techniques +de platification d’un module’. Invent Math 13 (1971), 1–89. +[Sai21] +S. Saito. ‘Reciprocity sheaves and logarithmic motives’. Preprint (2021). arXiv: +2107.00381. +[Sta22] +T. Stacks project authors. The Stacks project. https://stacks.math.columbia.edu. +2022. +[SV00] +A. Suslin and V. Voevodsky. ‘Relative Cycles and Chow Sheaves’. Cycles, trans- +fers, and motivic homology theories. Vol. 143. Ann. of Math. Stud. Princeton +Univ. Press, Princeton, NJ, 2000, 10–86. +23 + +Nikolai Opdan +[Voe00] +V. Voevodsky. ‘Triangulated Categories of Motives Over a Field.’ Cycles, trans- +fers, and motivic homology theories. Vol. 143. Ann. of Math. Stud. Princeton +Univ. Press, Princeton, NJ, 2000, 188–238. +[Voe10] +V. Voevodsky. ‘Cancellation theorem.’ Documenta Mathematica Extra Vol. +(2010), 671–685. +Department of Mathematics, University of Oslo, Moltke Moes vei 35, 0851 +Oslo, Norway +Email adress: +ntmarti@math.uio.no +URL: http://www.nikolaiopdan.com +24 + diff --git a/X9AzT4oBgHgl3EQfKfvG/content/tmp_files/load_file.txt b/X9AzT4oBgHgl3EQfKfvG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8b6c6226a04d82a308cf38a634d5c32c6d391f85 --- /dev/null +++ b/X9AzT4oBgHgl3EQfKfvG/content/tmp_files/load_file.txt @@ -0,0 +1,897 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf,len=896 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='01099v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='AG] 3 Jan 2023 LOGARITHMIC MOTIVES WITH COMPACT SUPPORT Nikolai Opdan Abstract We develop a theory of motives with compact support for logarithmic schemes over a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Starting from the notion of finite logarithmic correspondences with compact support, we define the logarithmic motive with compact support analogous to the classical case for schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We then establish an analog of a Gysin sequence and, assuming resolution of singularities, a Künneth formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This implies that our theory is □-invariant, which presents a critical feature that is absent in the classical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Further assuming resolution of singularities, we prove a duality theorem for log schemes which we use to establish a cancellation theorem for log schemes whose underlying scheme is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Moreover, we discuss new homology and cohomology theories for log smooth fs logarithmic schemes based on our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Introduction A theory of motives for logarithmic schemes has recently been developed by Binda–Park– Østvær [BPØ22a], [BPØ22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' One of their motivating ideas is to develop a theory of motives that captures information about a larger class of cohomology theories than those represent- able in Voevodsky’s triangulated category of motives DMeff(k, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Such theories include crystalline cohomology, Hodge cohomology, and de Rham–Witt cohomology [BPØ22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Des- pite not being A1-invariant, they satisfy a blow-up formula, projective bundle formula, Gysin sequence, and Mayer-Vietoris, which indicates that one should create a more general linear tensor category that also captures these theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Extending the theories to log schemes, one finds that they are insensitive to the log scheme □ := (P1, ∞), where the underlying scheme is the projective line and log structure given by the inclusion of the divisor ∞ ֒→ P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This indicates that one should create a theory of motives where □ takes the place of the unit object A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' However, since □ is a log scheme, one needs to develop a full motivic theory for log schemes to get a satisfactory framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Thus starts our quest for proving the fundamental properties of the triangulated category of logarithmic motives logDMeff(k, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' As a start, one expects to find analogs of classical results in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Indeed, Binda–Park–Østvær establishes a Gysin sequence, blow-up formula, projective bundle for- mula, and a Thom isomorphism for log schemes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' equipping schemes with a trivial log struc- ture, this recovers the classical results in algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Assuming resolutions of singu- larities, they moreover construct a fully faithful functor from DMeff(k, Λ) to logDMeff(k, Λ) and identify its essential image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' They also represent Hodge cohomology which shows that the functor is not essentially surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' These results enable one to view the new theory as an enlargement of the classical theory for log schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 1 Nikolai Opdan Voevodsky introduced in [Voe00] motives with compact support to state many essential results such as the localization sequence ([Voe00, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5]) and the duality theorem ([FV00, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2]) for non-proper schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Importantly, they also describe dual objects ([Voe00, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2]) and represent Bloch’s higher Chow groups for singular schemes ([Voe00, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The goal of this paper is to generalize this theory to logarithmic schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We begin by defining logarithmic correspondences with compact support for a perfect field k, and use this to define a presheaf with log transfers given by Y �→ Λc ltr(X)(Y ) := lCorc k(Y, X) ⊗Z Λ ∈ Pshltr(k, Λ) for any fs log scheme X and commutative ring Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' After establishing its basic properties, we show that its dividing Nisnevich sheafification a∗ dNisΛc ltr(X) is a dividing Nisnevich sheaf with log transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Its image in logDMeff(k, Λ) we call the logarithmic motive with compact support and we denote it by M c(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' It is contravariant for open embeddings and covariant for closed embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This construction resembles the classical construction of the motive with compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We show in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2 that M c(X) satisfies a Mayer–Vietoris property for coverings, that is, for every strict Nisnevich square in lSm/k Y ′ Y X′ X there is a homotopy cartesian square of log motives with compact support M c(Y ) M c(Y ′) M c(X) M c(X′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Moreover, for any log modification f : X → Y in lSm/k we establish an isomorphism M c(f) : M c(X) ≃ −→ M c(Y ) in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The classical theory of motives benefits greatly from the localization sequence [Voe00, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' In our framework, there can be no such sequence since it would imply A1-invariance, which is false since the essential image of the functor Rω∗ : DMeff(k, Λ) → logDMeff(k, Λ) consists of A1-local objects [BPØ22b, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='16], but is not essentially surjective since Ωj X/k is an object of logDMeff(k, Λ) for every X ∈ SmlSm/k [BPØ22b, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1] and the groups Hi Zar(X, Ωj X/k) ̸≃ Hi Zar(X × A1, Ωj X/k) are not isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We instead provide a distinguished triangle that will take play the role of the localization sequence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' that is, we prove a slight generalization of the following (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3): 2 Logarithmic motives with compact support – If Z is a smooth irreducible divisor on a smooth scheme X, there is a distinguished triangle M c(X, Z) → M c(X) → M c(Z)(1)[2] → M c(X, Z)[1] in logDMeff(k, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Assuming resolution of singularities, an important consequence is that the logarithmic motive with compact support factors over products, see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' – (Künneth formula) For every X, Y ∈ lSm/k there is an canonical map M c(X) ⊗ M c(Y ) −→ M c(X × Y ) which is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' As a corollary we get the logarithmic motive with compact support is □-invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=', – (Homotopy invariance) For every X ∈ lSm/k there is an isomorphism M c(X × □) ≃ M c(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This shows an important feature that differs from the classical theory since the classical motive with compact support is not homotopy invariant (A1-invariant) since M c(A1) ≃ Λ(1)[2] by [Voe00, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Having settled basic properties, we continue to work under the assumption of resolution of singularities, and relate our theory to the classical setting by establishing the isomorphism HomlogDMeff(k,Λ)(M(Y )[n], M c(X)) ≃ HomDMeff(k,Λ)(M(Y − ∂Y )[n], M c(X)) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This turns out to be helpful in proving a logarithmic analog of the duality theorem [FV00, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2] in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' – (Duality) For every X ∈ Sm/k and Y, T ∈ lSm/k such that T is of pure dimension d over k, there is an isomorphism Hom(M(Y × T )[n], M c(X)) ≃ Hom(M(Y × T )(d)[2d + n], M c(X × (T − ∂T ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We then prove of a logarithmic analog of the cancellation theorem [Voe10] for log schemes whose underlying scheme is proper in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' – (Cancellation) For every X, Y ∈ lSm/k such that the underlying schemes X, Y are proper over k, there is an isomorphism Hom(M(X), M(Y )) ≃ Hom(M(X)(1), M(Y )(1)) induced by tensoring with Λ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' It remains an open problem to prove the cancellation theorem for all fs log schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We end our discussion in section 5 by defining new homotopy invariants for log smooth fs log schemes which we call logarithmic cohomology with compact support defined by Hn,i lc (X, Λ) := HomlogDMeff(k,Λ)(M c(X), Λ(i)[n]) and logarithmic Borel–Moore homology defined by HlBM n,i (X, Λ) := HomlogDMeff(k,Λ)(Λ(i)[n], M c(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Based on our results, we show that they are contravariant for open immersions, covariant for closed immersions, has a Gysin triangle, and satisfies homotopy invariance and a duality statement (under the assumption of resolution of singularities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 3 Nikolai Opdan Outline – The paper is organized as follows: In section 2, we review the classical theory of motives with compact support, and we briefly discuss the logarithmic motivic theory from [BPØ22b] that will be used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We begin our theory in section 3 with a discussion on special correspondences needed to define the logarithmic motive with compact support in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This section presents the main body of the paper and is where we prove our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then, in section 5 we introduce new homology and cohomology theories for log schemes and prove some of their fundamental properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Notation and conventions – Fix a perfect field k, Λ an unital commutative ring, and let Sm/k denote the category of smooth and separated schemes of finite type over k and Psh(Sm/k, Λ) the category of presheaves of Λ-modules on Sm/k with coefficients in Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' By convention, all log schemes are separated and of finite type over Spec k, where Spec k is the point equipped with the trivial log structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The category of all fs log schemes we denote by lSch/k, while the category of all fs log smooth log schemes over k we denote by lSm/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' For a log scheme X we let X denote the underlying scheme, and SmlSm/k denote the full subcategory of lSm/k whose underlying scheme X is smooth over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' For a log scheme X we let ∂X denote the set of points on X with non-trivial log structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' If X is an fs log scheme, the complement of the log structure X −∂X will be an open subset of X, and there is a canonical open immersion X − ∂X → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We often assume that the field k admits (strong) resolution of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' By this, we mean that: for every integral scheme X of finite type over k, there exists a proper birational morphism Y → X over k such that Y is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' let f : Y → X be a proper birational morphism of integral schemes over k where X is smooth, and suppose that f −1(X − (Z1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' ∪ Zr)) → X − (Z1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' ∪ Zr) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then there is a sequence of blow-ups Xn fn−1 → Xn−1 fn−2 → · · · f0 → X0 = X along smooth centers Wi such that the composition Xn → X factors through f, the Wi are all contained in the preimage of Z1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' ∪ Zr in Xi, and the Wi have strict normal crossing with the sum of the strict transforms of Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' , Zr, f −1 0 (W0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' , f −1 i−1(Wi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Especially, this is satisfied if k has characteristic 0 by the work of Hironaka [Hir64] (see also [ATW19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Acknowledgments – I am grateful to my Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' advisor Paul Arne Østvær for his con- tinuous support and encouragement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' I wish to thank Doosung Park, Alberto Merici and Federico Binda for many helpful suggestions and fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The research is supported by the RCN Frontier Research Group project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 312472 “Equations in Motivic homotopy”, and I gratefully acknowledge the hospitality of the Uni- versità Degli Studi Di Milano and Centre for Advanced Study at the Norwegian Academy of Science and Letters while writing this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 4 Logarithmic motives with compact support 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Motives with compact support – Voevodsky developed a theory of motives with compact support in [Voe00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The basic idea is that there should be a theory that accompanies the theory of motives in the same way that cohomology with compact support accompanies ordinary cohomology theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Among other things, it was needed to state the localization theorem ([Voe00, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5]) and the duality theorem ([FV00, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2]) for non-proper schemes, and importantly it provided a description of the dualizing object, which for a smooth scheme X takes the form M(X)∗ := RHom(M(X), Λ(d))(−d) ≃ M c(X)(−d)[−2d] in DMgm(k, Λ) ([Voe00, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let Cork denote the category of finite correspondences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=', the objects are smooth schemes over k, and morphisms from X to Y are finite correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Recall the definition of the motive M(X) in DMeff(k, Λ) for a smooth scheme X over k as the image of the representable Nisnevish sheaf with transfers Λtr(−) : Cork → Shvtr Nis(Cork) in DMeff(k, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' For any U ∈ Sm/k, one defines Λtr(X)(U) as the free abelian group with coefficients in Λ of algebraic cycles on U ×X that are surjective and finite over a component of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' One defines similarly the motive with compact support M c(X) for any scheme X of finite type over k as the image in DMeff(k, Λ) of the sheaf with transfers zequi(−, r) : Cork → Shvtr Nis(Cork).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Here, for any for any U ∈ Sm/k, the group zequi(X, r)(U) is the free Λ-module of algebraic cycles on U ×X that are dominant and of relative dimension r over a component of U [SV00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' It can be checked that it is a Nisnevich sheaf with transfers, and that there is a natural inclusion Λtr(X) ֒→ zequi(X, 0) which is an equivalence if X is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The Nisnevich sheaf with transfers zequi(−, r) is covariant for closed maps and contravariant for étale morphisms in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' In general, if f : Y → X is a flat equidimensional morphism of relative dimension n, the flat pullback of cycles gives a morphism f ∗ : zequi(X, r) → zequi(Y, r + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Similarly, if g : Z → X is a proper equidimensional morphism of relative dimension n, the proper pushforward of cycles gives a morphism g∗ : zequi(Z, r) → zequi(X, n + r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' For any commutative ring Λ one defines motivic cohomology with compact support with coefficients in R by Hn,i c (X, Λ) := HomDMeff(k,Λ)(M c(X), Λ(i)[n], and dually, one defines Borel–Moore motivic homology by HBM n,i (X, Λ) := HomDMeff(k,Λ)(Λ(i)[n], M c(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 5 Nikolai Opdan For smooth schemes X, the fundamental comparison theorem ([Voe00, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='9]) gives an iso- morphism Hn,i(X, Z) ≃ −→ CHi(X, 2i − n) between motivic cohomology and Bloch’s higher Chow groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' For non-smooth schemes Y of dimension d, one instead have the isomorphism CHi(Y, n) ≃ HBM 2(d−i)+n,d−i(Y, Z), identifying higher Chow groups with Borel–Moore motivic homology groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' To summarize, the motive with compact support has the following properties: Properties 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' (Contravariant functoriality) For any flat map X → Y of relative dimension r there is a map f ∗ : M c(Y )(r)[2r] → M c(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' (Covariant functoriality) For any proper map of relative dimension 0 g : X → Y there is a map g∗ : M c(X) → M c(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' For every scheme X there is a canonical map M(X) → M c(X), which is an equivalence if X is proper over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' (Localization) Assume that k admits resolution of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' If Z i→ X is a closed subscheme with open complement U j→ X, then there is a distinguished triangle M c(Z) → M c(X) → M c(U) → M c(Z)[1] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1) ([Voe00, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' (Künneth formula) Assume that k admits resolution of singularities or the charac- teristic is invertible in the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then for any schemes X and Y there is an isomorphism M c(X × Y ) ≃ M c(X) ⊗tr L,Nis M c(Y ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2) ([Voe00, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' (Duality) Assume that k admits resolution of singularities or the characteristic is invertible in the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' If T is a smooth scheme of dimension d over k, then there are canonical isomorphisms Hom(M(X × T )[n], M c(Y )) ∼= Hom(M(X)(d)[2d + n], M c(Y × T ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3) for all n ∈ Z ([FV00, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' vii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' (Duality) Assume that k admits resolution of singularities or the characteristic is invertible in the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let Y → X be a flat equidimensional morphism of relative dimension n of finite type over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then there is a canonical morphism of the form f ∗ : M c(X)(n)[2n] → M c(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='4) (Projective bundle formula) If π : E → X is a rank n vector bundle, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='4) is an isomorphism ([Nie08, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 6 Logarithmic motives with compact support Moreover, [Voe00, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='8] shows that M c(An) ≃ Z(n)[2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2), we get for every scheme X an isomorphism M c(X × A1) ≃ M c(X)(1)[2] ̸≃ M c(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This shows that the motive with compact support is not homotopy invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' One of our goals is to create an analogue of the theory of motives with compact support for log schemes which is homotopy invariant, that is, if we let □ := (P1, ∞) parametrize our homotopies, the analogue of the motive with compact support for log schemes should satisfy the equivalence M c(X × □) ≃ M c(X) for all fs log schemes X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We will show this in corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Logarithmic motivic homotopy theory – Throughout this article, we work in the framework provided by [BPØ22b], which is the triangulated category of logarithmic motives over a field k with a trivial log structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' In this section, we give a brief construction and state basic properties that will be relevant for this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We will assume familiarity with the basic properties of logarithmic schemes, for which the standard reference is [Ogu18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let lSm/k denote the category of fine and saturated (abbreviated “fs”) log schemes that are log smooth over a field k with trivial log structure, and SmlSm/k the full subcategory consisting of fs log schemes X such that the underlying scheme X is smooth over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' For X, Y ∈ lSm/k, an elementary log correspondence Z from X to Y consists of an integral closed subscheme Z ⊂ X ×Y that is finite and surjective over a component of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' a morphism ZN → Y of fs log schemes, where ZN denotes the fs log scheme whose underlying scheme is the normalization of Z and the log structure MZN is given as the pullback p∗ logMX, where p : ZN → X denotes the induced scheme morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The category lCork of finite logarithmic correspondences over k is the category whose objects are fs log schemes that are log smooth over k and morphisms between two objects X and Y is the free abelian group generated lCork(X, Y ) generated by elementary log correspondences from X to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We let Λltr(X)(−) := lCork(−, X) ⊗ Λ denote the presheaf with transfers represented by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We consider two topologies on lSm/k: First, an analogue of the Nisnevich topology for log schemes called the strict Nisnevich topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' That is the Grothendieck topology associated to the cd-structure on lSm/k given by Y ′ Y X′ X, g′ f ′ f g 7 Nikolai Opdan where g is an open immersion, f is strict étale (a strict morphism of fs log schemes and an étale morphism of the underlying schemes), and f −1(X−g(X′)) → X−X′ is an isomorphism when both sides are considered with the reduced scheme structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Our prime examples for strict Nisnevich squares are the left and outer squares of the commutative diagram Gm AN A1 A1 □ P1, where AN := (A1, 0) and □ := (P1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Secondly, we consider the dividing topology: This is the Grothendieck topology asso- ciated to the cd-structure on lSm/k given by proper log étale surjective monomorphisms (or equivalently log modifications by ([BPØ22b, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='9])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The reason for considering this topology is that one wants to allow for blow-ups in the log structure without changing the topological structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We call the union of these two topologies the dividing Nisnevich topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Letting □ := (P1, ∞) be out replacement for A1, one defines the derived category of effective logarithmic motives logDMeff(k, Λ) as the homotopy category of C(Shvltr dNis(k, Λ)) with respect to the □-local descent model structure ([BPØ22b, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The logarithmic motive M(X) is the image of the sheafification for the dividing Nisnevich topology of Λltr(X) in logDMeff(k, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' By definition we have monoidal equivalences M(X × Y ) ≃ M(X) ⊗ M(Y ), and espe- cially □-homotopy invariance M(X × □) ≃ M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' There is a Mayer-Vietoris property for strict Nisnevich squares, and every log modification f : Y → X induces an equivalence of log motives M(f) : M(Y ) ≃ M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' It is shown in [BPØ22b] that there are generaliz- ation to log schemes of the classical Gysin triangles ([BPØ22b, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='4]), Blow-up triangles ([BPØ22b, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3]), the Projective bundle theorem ([BPØ22b, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5]), and the Thom iso- morphism ([BPØ22b, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Removing the log structure, there is a functor ω : lSm/k → Sm/k that sends an object X to X −∂X, where ∂X denotes the set of points on X with nontrivial log structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Assuming that k admits resolution of singularities, this functor induces an adjoint functor pair ω♯ : logDMeff(k, Λ) DMeff(k, Λ) : Rω∗, and Rω∗ is shown to be fully faithful in [BPØ22b, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Identifying the essential image of Rω∗ in logDMeff(k, Λ) one finds the smallest subcategory that is closed under small sums and shifts and contains every motive M(X) where X ∈ lSm/k is proper over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' How- ever, since the sheaf of logarithmic differentials Ωj /k are representable in logDMeff(k, Λ) ([BPØ22b, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1]), but not in DMeff(k, Λ), this shows that Rω∗ is not essentially surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 8 Logarithmic motives with compact support We briefly mention the paper [BM21] which shows that logDMeff(k, Λ) admits a t- structure which induces the t-structure on DMeff(k, Λ) under ω♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' They generalize the clas- sical purity theorem and Morel’s connectivity theorem to the logarithmic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' However, we will not make use of any of these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Logarithmic correspondences with compact support 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Finite logarithmic correspondences with compact support – To introduce the logarithmic motive with compact support we follow the classical approach by first introduce logarithmic correspondences with compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' For X ∈ lSm/k and Y ∈ lSch/k, an elementary logarithmic correspondence with compact support Z from X to Y is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' an integral closed subscheme Z ⊂ X×Y such that the projection Z → X is quasi-finite and dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' a morphism ZN → Y of fs log schemes, where ZN denotes the fs log scheme whose underlying scheme is the normalization of Z, and if p : ZN → X denotes the induced scheme morphism, then the log structure MZN is given as p∗ MX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' For any ring Λ, the group lCorc k(X, Y ) ⊗Z Λ is the free abelian Λ-module generated by elementary logarithmic correspondence with compact support between X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This definition is a rephrasing of the definition of correspondences with com- pact support given in [SV00] (see also [Blo86]) adapted to logarithmic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The first condition is a direct translation of the standard definition, while the second condition is necessary to construct compositions of log correspondences with compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We note that the second condition is equivalent to the second condition in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1 in [BPØ22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Some properties are immediate: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Since the underlying scheme Z of a log correspondence Z is finite and surjective over a component of X, we have lCork(X, Y ) ⊂ lCorc k(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' If Y has trivial log structure, then the second condition is trivially satisfied, and lCorc k(X, Y ) ≃ zequi(Y, 0)(X), where zequi(−, r) is sheaf of relative dimension r cycles as defined in [SV00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' If X has trivial log structure, then ZN has trivial log structure, and ZN → Y factors through Zn → Y − ∂Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Hence lCorc k(X, Y ) ≃ Corc k(X, Y − ∂Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' If both X and Y has trivial log structure, then combining the two above properties we get lCorc k(X, Y ) ≃ Corc k(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 9 Nikolai Opdan Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' For X ∈ lSch/k we define a presheaf of Λ-modules with transfers Λc ltr(X) on lCork by Y �→ Λc ltr(X)(Y ) := lCorc k(Y, X) ⊗ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Given a proper morphism f : S′ → S, the pullback of cycles induces a natural map f ∗ : Λc ltr(X)(S) → Λc ltr(X)(S′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' On the other hand, a flat morphism g : S′ → S of relative dimension 0 (an open immersion) induces a map g∗ : Λc ltr(X)(S) → Λc ltr(X)(S′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The injection Λltr(X)(−) ֒→ Λc ltr(X)(−) makes the representable sheaf Λltr(X) a sub- sheaf of Λc ltr(X) for all X since the structure morphism associated to U → V is compatible with log transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' If p : ZN → X and q : ZN → Y denotes the underlying scheme morphisms, then referring to Definition III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5 in [Ogu18], the second condition of definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1 is equivalent to the giving a morphism q∗ log MY → p∗ log MX [BPØ22b, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' From this, we see that Λc ltr(ptN) is the empty sheaf since there can be no such morphism q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let X be an fs log scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' If X is proper, then Λc ltr(X) = Λltr(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' If X is proper, the underlying scheme X is proper as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This implies that the underlying scheme Z of a finite log correspondence Z ∈ Λc ltr(X)(U) is proper over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Since Z now is both quasi-finite and proper, [Sta22, 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1] shows that it is also finite over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We conclude that Z ∈ Λltr(X)(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' A coherent log scheme X over k is said to be solid if for any point x ∈ X the induced map Spec (OX,x) → Spec (MX,x) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let f : X → Y be a quasi-finite and dominant morphism of fs log schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' If Y is smooth over k, then X is solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Owing to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='7 in [BPØ22b] it suffices to show that f is an open immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Since Y is normal by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='8 in [BPØ22b], we conclude that f is universally open by Lemma 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2 in [Sta22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let X, Y ∈ lSm/k and suppose that Z is a closed subscheme of X × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then there exists at most one elementary log correspondence with compact support whose under- lying scheme is Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Using lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='7, this follows the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1 in [BPØ22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 10 Logarithmic motives with compact support Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let X and Y be fs log schemes that are log smooth over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then the homo- morphism of abelian groups lCorc k(X, Y ) −→ lCorc k(X − ∂X, Y − ∂Y ), defined by V �→ V − ∂V , is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Using lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='8 this follows the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2 in [BPØ22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let X be an fs log scheme log smooth over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then Λc ltr(X) is a strict étale sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This is essentially the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1 in [BPØ22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' For every X ∈ lSm/k, the sheaf a∗ dNisΛc ltr(X) is a dividing Nisnevish sheaf with log transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Since Λc ltr(X) is a strict étale sheaf by proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='10, we may apply the dividing Nisnevich sheafication functor a∗ dNis : Pshlog(Sm/k) −→ Shvlog(Sm/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The result now follows from [BPØ22b, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='7] which states that the dividing Nisnevich topo- logy is compatible with log transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' [BPØ22b, Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1] An admissible blow-up is a proper birational map X′ → X of fs log schemes over k such that the induces morphism X′ − ∂X′ → X − ∂X is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We let ABl /k denote the class of admissible blow ups, and we let (ABl /k) � Y denote the class of all admissible blow-ups over Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Assume that k admits resolution of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let X be a smooth scheme over k and Y an fs log scheme log smooth over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then there is a naturally induced isomorphism colim Y ′∈(ABl /k)�Y Λc ltr(X)(Y ′) ≃ Λc ltr(X)(Y − ∂Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Following the proof of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1 [BPØ22b], we let Y ′ → Y be an admissible blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The induced isomorphism Y ′ − ∂Y ′ → Y − ∂Y allows for the construction of a the composite homomorphism ϕY ′ : Λc ltr(X)(Y ′) → Λc ltr(X)(Y ′ − ∂Y ) ≃ Λc ltr(X)(Y − ∂Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Gathering the ϕY ′ we get a morphism ϕ : colim Y ′∈(ABl /k)�Y Λc ltr(X)(Y ′) → Λc ltr(X)(Y − ∂Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let W ∈ Λc ltr(X, Y ′) be an elementary log correspondence with compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' If ϕY ′(W) = 0, then W = 0, since W is the closure of ϕY ′(W) ∈ Y ′ × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 11 Nikolai Opdan For the surjectivity, let Z ∈ Λc ltr(X, Y − ∂Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' By Raynaud–Gruson platification ([RG71, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2]), there is a scheme Y ′ and a morphism f : Y ′ → Y such that f is an isomorphism on Y − ∂Y and the closure of Z′ of Z ⊂ Y ′ × X is flat over Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The subscheme Z′ is a correspondence with compact support from Y ′ to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' By resolution of singularities there is a blow up g : Y ′′ → Y ′ such that Y ′′ is smooth over k, and the complement of (f ◦ g)−1(Y − ∂Y ) in Y ′′ consists of strict normal crossing divisors Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' , Zr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Letting Y ′′ = (Y ′′, Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' + Zr), the induced morphism Y ′′ → Y of fs log schemes is an admissible blow-up, and the closure Z′′ of Z in Y ′′ × X is a closed subscheme of Z′ ×Y ′ Y ′′, which preserved under base change, is quasi-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' It follows that Z can be extended to a finite correspondence with compact support from Y ′′ to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Since X has trivial log structure, this gives a log correspondence with compact support from Y ′′ to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Thus ϕ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' If f ′ : X′ → X is a log modification in lSm/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' For every Y ∈ lSm/k and V ∈ lCorc k(Y, X) there exists a dividing Zariski cover g′ : Y ′ → Y and W ∈ lCorc k(Y ′, X′) fitting in a commutative diagram Y ′ X′ Y X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' W g f V Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Similar to the proof for Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5 in [BPØ22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Recall from [BPØ22b] the adjuction of categories Pshlog(k, λ) Pshltr(k, λ) Shvlog(k, λ) Shvltr(k, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' γ♯ a∗ dNis γ∗ a∗ dNis γ♯ adNis ∗ γ∗ adNis ∗ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Suppose that f : Y → X is an log modification of fs log schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then the induced morphism a∗ dNisγ∗Λc ltr(Y ) −→ a∗ dNisγ∗Λc ltr(X) of dividing Nisnevich sheaves is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' For any fs log scheme T there is a commutative diagram Λc ltr(Y )(T ) Λc ltr(X)(T ) Λc tr(Y − ∂Y )(T − ∂T ) Λc tr(X − ∂X)(T − ∂T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The vertical morphisms are injections by lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='9, and since the lower morphism is an isomorphism since Y − ∂Y ≃ X − ∂X, the top morphism is injective as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Hence Λc ltr(Y ) → Λc ltr(X) is a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This implies that a∗ dNisγ∗Λc ltr(Y ) −→ a∗ dNisγ∗Λc ltr(X) 12 Logarithmic motives with compact support is a monomorphism since a∗ dNis and γ∗ are exact by [BPØ22b, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Due to lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='14 there exists a dividing Nisnevich cover g : T ′ → T and a finite log correspondence with compact support W ∈ Λc ltr(Y )(T ′) such that f ◦ W = g ◦ V for every V ∈ Λc ltr(Y )(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This proves that a∗ dNisΛc ltr(Y ) −→ a∗ dNisΛc ltr(X) is an epimorphism, which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Logarithmic motives with compact support 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Definition of logarithmic motives with compact support – In this section, we introduce the logarithmic motive with compact support and establish basic properties using the results above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' For any X ∈ lSm/k, the logarithmic motive with compact support of X, denoted M c(X), is the image of a∗ dNisΛc ltr(X) in logDMeff(k, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Given a morphism of fs log schemes f : Y → X, we let M c(Y f→ X) denote the cone in logDMeff(k, Λ) associated with the morphism of complexes a∗ dNisΛc ltr(Y ) → a∗ dNisΛc ltr(X) in C(Shvltr dNis(k, Λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We also denote it by M c(Y → X) if the morphism is understood, or sometimes simply M c(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The inclusion Λltr(X) ֒→ Λc ltr(X) induces a canonical map M(X) ֒→ M c(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' If X is proper, proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5 implies that this map is an isomorphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' we have M c(X) ≃ M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1) Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Suppose that Y ′ Y X′ Y is a dividing Nisnevich distinguished square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then there is an induced isomorphism M c(Y → X) ≃ −→ M c(Y ′ → X′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We need to show that for every strict □-invariant complex F of dividing Nisnevich sheaves with log transfers there is an isomorphism HomlogDMeff(k,Λ)(M c(Y → X), F) −→ HomlogDAeff(k,Λ)(M c(Y ′ → X′), F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We may further assume that F is a fibrant object in C(Shvltr dNis(k, Λ)) with regard to the □-local descent structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' By arguing similar to [BPØ22b, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3] this reduces the problem to showing that there is an isomorphism HomD(Shvltr dNis(k,Λ))(a∗ dNisΛc ltr(Y → X), F) ≃ HomD(Shvltr dNis(k,Λ))(a∗ dNisΛc ltr(Y ′ → X′), F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 13 Nikolai Opdan The result now follows if we can show that the exactness of the sequence of dividing Nisnevich sheaves with log transfers 0 → a∗ dNisΛc ltr(Y ) f→ a∗ dNisΛc ltr(Y ′) ⊕ a∗ dNisΛc ltr(X) g→ a∗ dNisΛc ltr(X′) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2) For [BPØ22b, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5] the sequence 0 → a∗ dNisΛltr(Y ) → a∗ dNisΛltr(Y ′) ⊕ a∗ dNisΛltr(X)→a∗ dNisΛltr(X′) → 0 is exact, and using the canonical inclusions Λltr(X) → Λc ltr(X) we get a commutative dia- gram 0 a∗ dNisΛc ltr(Y ) a∗ dNisΛc ltr(Y ′) ⊕ a∗ dNisΛc ltr(X) a∗ dNisΛc ltr(X′) 0 0 a∗ dNisΛltr(Y ) a∗ dNisΛltr(Y ′) ⊕ a∗ dNisΛltr(X) a∗ dNisΛltr(X′) 0, f g which proves the injectivity of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The exactness at a∗ dNisΛc ltr(Y ′)⊕a∗ dNisΛc ltr(X) follows from the commutativity of the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' For the surjectivity of g we choose compactifications X′ → X′, Y ′ → Y ′, X → X, and y → Y and get a commutative diagram 0 a∗ dNisΛltr(Y ) a∗ dNisΛc ltr(Y ′) ⊕ a∗ dNisΛc ltr(X) a∗ dNisΛc ltr(X) 0 0 a∗ dNisΛc ltr(Y ) a∗ dNisΛc ltr(Y ′) ⊕ a∗ dNisΛc ltr(X) a∗ dNisΛc ltr(X′) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' f g The surjectivity of g follows from the vertical morphisms’ surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This proves the exactness of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2) which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Gysin triangles – In [Voe00, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='4] Voevodsky constructed a Gysin triangle for mixed motives over a perfect field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' That is, for a closed isomorphism i : Z → X between smooth k-schemes, there is a distinguished triangle M(X − Z) → M(X) i∗ → M(Z)(n)[2n] → M(X − Z)[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This triangle corresponds to the localization sequence in cohomology, which is an important tool in studying higher Chow groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This section aims to prove a weak analog for log schemes that generalize parts of the classical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Namely, we will prove the following: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let X be a smooth scheme over k and suppose that Z1, Z2, · · · , Zr are smooth irreducible divisors forming a strict normal crossing divisor Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' + Zr on X ∈ Sm/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let Zs,t := Zs + Zs+1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' + Zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then there exists a distinguished triangle M c(X, Z1,r) M c(X, Z1,r−1) M c(Zr, Zr ∩ Z1,r−1)(1)[2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' +1 in logDMeff(k, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 14 Logarithmic motives with compact support Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Suppose that Z is a smooth irreducible divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Applying the technique of deformation to the normal cone in [BPØ22b, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='8], originally due to Fulton [Ful84], we Nisnevich locally on X have a Cartesian diagram Z X As As+1, u where u is an étale morphism and s denotes the dimension of Z over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let X1 = As Z and ∆ be the diagonal of Z over As, and let X2 := X ×As X1 − (X ×As +1 Z − ∆) ∪ (Z ×As+1 X1 − ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let DZX denote the deformation space (Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1 in [BPØ22b]) of the pair (X, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Arguing similar to [BPØ22b, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='4], since blow-ups commute with flat base change, we have a Cartesian square DZX2 X2 × □ DZX X × □ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This implies that the map DZX2 → DZX is étale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Furthermore, since X2 → X and NZX2 → NZX are also étale, where NZX is the normal bundle of Z in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Strict Nisnevich descent then implies that there are isomorphisms M c((BlZX) → X) ≃ −→ M c((BlZX2, E2) → X2) M c((BlZ×□(DZX)) → DZX) ≃ −→ M c((BlZ×□(DZX2), ED 2 ) → DZX2) M c((BlZ(NZX)) → NZX) ≃ −→ M c((BlZ(NZX2), EN 2 ) → NZX2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Applying deformation to the normal cone, we can replace X by X2, and similarly, we can repeat the argument to replace X2 by X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This reduces this case to proving it for X = A1 and Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Using the square (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2) and strict Nisnevich descent (proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2) we have isomorph- isms M c((Bl0P1, E) → P1) ≃ −→ M c((Bl0 A1, E) → A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Invoking Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='4 in [BPØ22b] we get the distinguished triangle M c(Bl0P1, E) → M c(P1) → Λ(1)[2] → M c(Bl0P1, E), hence the distinguished triangle M c(Bl0 A1, E) → M c(A1) → Λ(1)[2] → M c(Bl0 A1, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Applying dividing descent (proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2) to the log modification (Bl0 A1, E) → (A1, 0) 15 Nikolai Opdan we have the desired triangle M c(A1, 0) → M c(A1) → M c(0)(1)[2] → M c(A1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' If Z = Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' + Zr with each Zi is irreducible, we by induction we a diagram M c(X, Z1,r) M c(X, Z2,r) M c(Z1, Z1 ∩ Z2,r)(1)[2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' M c(X, Z1,r−1) M c(X, Z2,r−1) M c(Z1, Z1 ∩ Z2,r−1)(1)[2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' M c(Zr, Zr ∩ Z1,r−1)(1)[2] M c(Zr, Zr ∩ Z2,r−1)(1)[2] M c(Z1 ∩ Zr)(2)[4] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' +1 +1 +1 +1 +1 +1 where each square is commutative, and where the two bottom rows and two right columns are distinguished triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then the 3×3-lemma ([May01, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='6]) assures that the top and left triangles are cofibre sequences as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let Z be a smooth irreducible divisor on a smooth scheme X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then there exists a distinguished triangle M c(X, Z) → M c(X) → M c(Z)(1)[2] → M c(X, Z)[1] in logDMeff(k, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Consider the open immersion A1 ֒→ □.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' One cannot yet state a localization sequence analogous to the classical case (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' After all, it would involve the definition of the motive of a point with a nontrivial log structure, which is not defined because it is not log smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' In lack of an analogy for this important theorem, we will often instead be able to use the above corollary and the general theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Künneth formula – Classically, the Künneth formula [Voe00, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='7] for motives with compact support is a direct consequence of the localization theorem [Voe00, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' However, we prove it directly since we do not yet have such a result (see remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let v : lCork → lCork[(ABl /k)−1] denote the localization functor where ABl /k denotes the class of admissible blow-ups that were defined in definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then there exists functors C≤0(Psh(lCork, Λ)) C≤0(Psh(lCork[(ABl /k)−1], Λ)) v♯ v∗ v∗ where v♯ is left adjoint to v∗ and v∗ is left adjoint to v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Assume that k admits resolution of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' For every smooth scheme X over k, there is an isomorphism a∗ dNisΛc ltr(X) ≃ ω∗Λc tr(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 16 Logarithmic motives with compact support Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3 in [BPØ22b] there is an isomorphism a∗ dNisv∗v♯Λc ltr(X) ≃ v∗v♯a∗ dNisΛc tr(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Applying proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='13 and using the identification (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1) in [BPØ22b] we get an iso- morphism a∗ dNisv∗v♯Λc ltr(X) ≃ ω∗a∗ dNisΛc ltr(X), and using Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='8 in [BPØ22b] we have v∗v♯a∗ dNisΛc ltr(X) ≃ a∗ dNisΛc ltr(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We conclude by the face that a∗ dNisΛc ltr(X) is a dividing Nisnevich sheaf, since the functor η♯ : Shv(két, Λ) → Shvl ét(k, Λ) is fully faithful (Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2 in [BPØ22b]), where η : két → lSm/k is the inclusion functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Recall from [BPØ22b, 156] the definition of the complex C∗ F for a presheaves with log transfers F and U ∈ lSm/k as C∗ F(U) := � · · → colim (Y ∈ABl)�(U×∆ 1 □ ) F(Y ) → colim (Y ∈ABl)�U F(Y ) � Similarly, if F is bounded above we define the double Suslin complex for U ∈ Sm/k as CA1 ∗ F(U) := � · · → F(U × ∆1 A1) → F(U) � where ∆n A1 := Spec k[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' , xn]/ � n � i=0 xi = 1 � is the standard A1-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Assume that k admits resolution of singularities and let X be a smooth scheme over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then there are isomorphisms ω∗Λc tr(X) ≃ C∗ω∗Λc XCA1 ∗ ω∗Λc X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The first isomorphism comes from Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='4(4) and Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='8 in [BPØ22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The second isomorphism comes from applying [BPØ22b, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The following result is a generalization of [BPØ22b, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='6] to non-proper schemes X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Assume that k admits resolution of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let X ∈ Sm/k and Y ∈ lSm/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then for any n ∈ Z there is an isomorphism HomlogDMeff(k,Λ)(M(Y )[n], M c(X)) ≃ HomDMeff(k,Λ)(M(Y − ∂Y )[n], M c(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 17 Nikolai Opdan Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='8 gives an equivalence between HomlogDMeff(k,Λ)(M(Y )[n]), ω∗Λc tr(X)) and HomlogDMeff(k,Λ)(M(Y )[n]), ω∗CA1 ∗ Λc tr(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The complex CA1 ∗ Λc tr(X) is strictly A1-invariant, hence it is strictly □-invariant in the divid- ing Nisnevich topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Applying Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3 in [BPØ22b] there is an isomorphism HomlogDMeff(k,Λ)(M(Y ), ω∗CA1 ∗ Λc tr(X)[n]) ≃ Hn dNis(Y, ω∗CA1 ∗ Λc tr(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Further applying Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='10 in [BPØ22b] we have Hn dNis(Y, ω∗CA1 ∗ Λc tr(X)) ≃ Hn Nis(Y − ∂Y, CA1 ∗ Λc tr(X)), which by Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='16 gives an isomorphism Hn Nis(Y − ∂Y, CA1 ∗ Λc tr(X)) ≃ HomDMeff(k,Λ)(M(Y − ∂Y ), M c(X)[n]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Moving the shift gives the right isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Assume that k admits resolution of singularities, and let X be a smooth scheme over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then the unit of adjunction id → Rω∗ω♯ induces an equivalence M c(X) ≃ Rω∗ω♯M c(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' It suffices to show that for every generator M(Y )[i] for Y ∈ lSm/k and i ∈ Z that there is an isomorphism Hom(M(Y )[i], M(X)) ≃ Hom(M(Y )[i], Rω∗M(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Using the isomorphisms Hom(M(Y )[i], Rω∗M(X)) ≃ HomDMeff(k,Λ)(ω♯M(Y )[i], M(X)) ≃ HomDMeff(k,Λ)(M(Y − ∂Y )[i], M(X)), the result follows from theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Taking X = A1 in the corollary we get the isomorphisms M c(A1) ≃ Rω∗ω♯M c(A1) ≃ Rω∗M c(A1) ≃ Rω∗Λ(1)[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This gives the calculation M c(A1) ≃ Λ(1)[2] (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3) in logDMeff(k, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Assume that k admits resolution of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let X and Y be log smooth fs log schemes over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then there is an canonical equivalence M c(X) ⊗ M c(Y ) ≃ −→ M c(X × Y ) in logDMeff(k, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 18 Logarithmic motives with compact support Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Step 1: Assuming resolution of singularities, and for X, Y ∈ Sm/k, we have ω∗(M c(X)) ⊗ ω∗M c(Y )) ≃ ω∗M c(X × Y ) by the classical Künneth formula ([Voe00, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The result then follows by applying Rω∗ and using Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' If X and Y are proper fs log schemes it follows from the monoidal structure in logDMeff(k, Λ) using that M c(X) = M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Step 2: Assume that X ∈ SmlSm/k and Y ∈ Sm/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' By [BPØ22b, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='10] we can assume that X is of the form (X, Z), where Z is a strict normal crossing divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Writing Z = Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Zr we use induction on r to find a diagram of cofibre sequences M c(X, Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' + Zr) ⊗ M c(Y ) M c(X × Y, (Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' + Zr) × Y ) M c(X, Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' + Zr−1) ⊗ M c(Y ) M c(X × Y, (Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' + Zr−1) × Y ) M c(Zr, Zr ∩ (Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' + Zr−1))(1)[2] ⊗ M c(Y ) M c(Zr × Y, (Zr ∩ (Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' + Zr)) × Y )(1)[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The induction hypothesis shows that the two lower horizontal maps are isomorphisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' hence we conclude that the top morphism is also an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Step 3: Assume that X ∈ lSm/k and Y ∈ Sm/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then by toric deformation ([BPØ22b, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2]) and proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2 we reduce to Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' By symmetry, we can apply the arguments to Y as well when X ∈ Sm/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Step 4: It remains to show the general case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=', for X, Y ∈ lSm/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Writing Z = Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' + Zr and W = W1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' + Ws we find that applying theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3 twice gives us a commutative diagram M c(X, Z1,r) ⊗ M c(Y, W1,s) M c(X × Y, Z1,r × Y + W1,s × X) M c(X, Z1,r−1) ⊗ M c(Y, W1,s−1) M c(X × Y, Z1,r−1 × Y + W1,s−1 × X) M c(Zr, Zr ∩ Z1,r−1)(1)[2] ⊗ M c(Ws, Ws ∩ W1,s−1)(1)[2] M c(Zr × Ws, (Zr ∩ Z1,r−1) × Ws + (Ws ∩ W1,s−1) × Zr)(2)[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' By double induction on r and s, we find that the two lower horizontal morphisms are isomorphisms, proving that the top horizontal morphism is also an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Step 2 in the proof is essential as it proves homotopy invariance of the log motive with compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' This marks a significant differentiation from the classical theory, which is not homotopy invariant (A1-invariant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Using calculations in [BPØ22b], we make some easy computations in the following corol- lary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Assume that k admits resolution of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then for any X ∈ lSm/k there are isomorphisms M c(X × □) ∼= M c(X), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='4) M c(X × (Pn+1, Pn)) ∼= M c(X), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5) and M c(X × Pn) ∼= M c(X) ⊗ n � i=0 Λ(i)[2i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='6) 19 Nikolai Opdan Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The first and second isomorphisms uses □-invariance and (Pn+1, Pn) of the log motive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The final isomorphism follows from the calculation M(Pn) ≃ n � i=0 Λ(i)[2i] from Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='6 in [BPØ22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Duality – One uses the motive with compact support to state a duality theorem for non-proper schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We will establish the same statement for fs log schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Although the motive of A1 is unknown, it follows from work of Saito that the category of reciprocity sheaves embeds fully faithfully in logDMeff(k, Λ) ([Sai21]), that the relations between the Witt-vectors and M(A1) forces M(A1) to be very big.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Using that M c(A1) is merely Z(1)[2], this gives an affirmative answer to the question raised in [BPØ22b, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='7] that the equivalence HomlogDMeff(k,Λ)(M(Y )[i], M(X)) ≃ HomDMeff(k,Λ)(M(Y − ∂Y )[i], M(X)) does not necessarily hold for non-proper schemes X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' One then has the analogue of the duality theorem [FV00, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2] for fs log schemes: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Assume that k admits resolution of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let X ∈ Sm/k and Y, T ∈ lSm/k such that T is of pure dimension d over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then for any n ∈ Z there is an isomorphism Hom(M(Y × T )[n]), M c(X)) ≃ Hom(M(Y )(d)[2d + n]), M c(X × (T − ∂T ))) in logDMeff(k, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' By theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='9 there is a isomorphism between HomlogDMeff(k,Λ)(M(Y × T )[n], M c(X)) and HomDMeff(k,Λ)(M((Y − ∂Y ) × (T − ∂T ))[n], M c(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Using the classical duality theorem ([FV00, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2]) by Friedlander and Voevodsky, there is further an isomorphism to HomDMeff(k,Λ)(M(Y − ∂Y )(d)[2d + n], M c(X × (T − ∂T ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Reapplying theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='9 we get an isomorphism to HomlogDMeff(k,Λ)(M(Y )(d)[2d + n], M c(X × (T − ∂T ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The composition of these isomorphisms give the desired isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 20 Logarithmic motives with compact support 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Cancellation – Cancellation is classically known as the equivalence HomDMeff(k,Λ)(M, N) ≃ HomDMeff(k,Λ)(M(1), N(1)) induced by tensoring with Λ(1), also known as Gm-cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The fact that this is true for perfect fields k is a difficult theorem due to Voevodsky in [Voe10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' However, if one assumes resolution of singularities, a much simpler proof is available using the motive with compact support ([Voe00, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Assuming resolution of singularities, the cancellation property in the full subcategory logDMeff prop(k, Λ) of logDMeff(k, Λ) generated by fs log schemes whose underlying scheme is proper over k follows the fact that the Rω∗ : DMeff(k, Λ) −→ logDMeff(k, Λ) is fully faithful by [BPØ22b, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='13] and from the identification of its essential image as logDMeff prop(k, Λ) by [BPØ22b, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We will instead combine almost all our above results to give an alternative easier proof without relying on these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Assume that k admits resolution of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let M and N be two objects of logDMeff prop(k, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then tensoring with Λ(1) induces an isomorphism Hom(M, N) ≃ Hom(M(1), N(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We reduce to the case when M, N ∈ SmlSm/k and begins by assuming that N = M(X, Z) where X is a smooth scheme over k and Z is a smooth irreducible divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Then corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='4 gives two cofiber sequences sitting in a commutative square Hom(M, M(X, Z)) Hom(M(1), M(X, Z)(1)) Hom(M, M(X)) Hom(M(1), M(X)(1)) Hom(M, M(Z)(1)[2]) Hom(M(1), M(Z)(2)[2]), where the two lower horizontal injections are the result of the composition of the morphisms Hom(M(Y ), M(X)) ֒→ Hom(M(Y ), M c(X)) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='9 ≃ HomDMeff(k,Λ)(M(Y − ∂Y ), M(X)) ≃ HomDMeff(k,Λ)(M(Y − ∂Y )(1), M c(X)(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' These maps are all isomorphisms if X is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Applying induction to the number of irreducible components of Z = Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' + Zr, we use theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='3 twice to get a commutative diagram 21 Nikolai Opdan Hom(M, M(X, Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Zr)) Hom(M(1), M(X, Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Zr)(1)) Hom(M, M(X, Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Zr−1)) Hom(M(1), M(X, Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Zr−1)(1)) Hom(M, M(Zr, Zr ∩ (Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Zr−1))(1)[2]) Hom(M(1), M(Zr, Zr ∩ (Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Zr−1))(2)[2])), The induction hypothesis implies that the two bottom horizontal morphisms are iso- morphisms, which also shows that the top morphism is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Question 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Does the cancellation theorem hold in greater generality in logDMeff(k, Λ)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Logarithmic invariants with compact support 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Logarithmic Borel–Moore homology and logarithmic motivic cohomology with compact support – Having defined the logarithmic motive with compact support and worked out some properties, we derive new invariants for logarithmic schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' For any fs log scheme X and any ring Λ, we define logarithmic Borel–Moore homology by HlBM n,i (X, Λ) := HomlogDMeff(k,Λ)(Λ(n)[i], M c(X)), and logarithmic motivic cohomology with compact support by Hn,i lc (X, Λ) := HomlogDMeff(k,Λ)(M c(X), Λ(n)[i]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Because of the results from the previous sections, we easily deduce some properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' We state these for logarithmic Borel–Moore homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Properties 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Let X, Y, Z be fs log schemes log smooth over k and n, i be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' (Contravariant functoriality) For any flat morphism f : X → Y of relative dimension 0 we have an induced morphism f ∗ : HlBM n,i (Y, Λ) → HlBM n,i (X, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' (Covariant functoriality) For any proper map g : Z → Y of relative dimension 0 there is a map g∗ : HlBM n,i (Z, Λ) → HlBM n,i (Y, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' (Homotopy invariance) Assume that k admits resolution of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' For any X there is an isomorphism HlBM n,i (X × □, Λ) ≃ HlBM n,i (X, Λ) by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' (Weak Gysin triangle) If V is a smooth irreducible divisor in X we find by applying Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='4 that there is a distinguished triangle HlBM n,i ((X, Z), Λ) → HlBM n,i (X, Λ) → HlBM n+2,i+1(Z, Λ) → HlBM n+1,i((X, Z), Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='1) 22 Logarithmic motives with compact support v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Assume that k admits resolution of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' If W is a smooth scheme and j is a positive integer we get an isomorphism HlBM n,i (W, Λ) ≃ HlBM n+j,i+2j(W, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='2) by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='15.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Preprint (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' arXiv: 2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='00381.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' [Sta22] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Stacks project authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' The Stacks project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' https://stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' [SV00] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Suslin and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Voevodsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' ‘Relative Cycles and Chow Sheaves’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Cycles, trans- fers, and motivic homology theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Princeton Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Press, Princeton, NJ, 2000, 10–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 23 Nikolai Opdan [Voe00] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Voevodsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' ‘Triangulated Categories of Motives Over a Field.’ Cycles, trans- fers, and motivic homology theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Princeton Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Press, Princeton, NJ, 2000, 188–238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' [Voe10] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Voevodsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' ‘Cancellation theorem.’ Documenta Mathematica Extra Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' (2010), 671–685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content=' Department of Mathematics, University of Oslo, Moltke Moes vei 35, 0851 Oslo, Norway Email adress: ntmarti@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='uio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='no URL: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='nikolaiopdan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} +page_content='com 24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfKfvG/content/2301.01099v1.pdf'} diff --git a/XNE5T4oBgHgl3EQfcw99/content/2301.05606v1.pdf 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Our method +is based on the state-of-the-art Natural Language Processing (NLP) building blocks, combined +with a novel technique for infusing the meta-data input that is typically available in messages +such as the sender information, timestamps, attached image, audio, affiliations, and more. As we +demonstrate throughout the paper, going beyond the mere text by leveraging all available channels +in the message, could yield an improved representation and higher classification accuracy. To +achieve message representation, each type of input is processed in a dedicated block in the neural +network architecture that is suitable for the data type. Such an implementation enables training all +blocks together simultaneously, and forming cross channels features in the network. We show in the +Experiments Section that in some cases, message’s meta-data holds an additional information that +cannot be extracted just from the text, and when using this information we achieve better performance. +Furthermore, we demonstrate that our multi-modality block approach outperforms other approaches +for injecting the meta data to the the text classifier. +Keywords Message classification · Meta data injection · Deep learning · Natural language processing +1 +Introduction +Many real world applications require message classification and regression, such as handling spam emails [1], ticket +routing [2], article sentiment review [3] and more. Accurate message classification could improve critical scenarios +such as in call centers (routing tickets based on topic) [2], alert systems (flagging highly important alert messages) [4], +and categorizing incoming messages (automatically unclutter emails) [1, 5]. The main distinction between text and +message classification is the availability of additional attributes, such as the sender information, timestamps, attached +image, audio, affiliations, and more. New message classification contests often appear in the prominent platforms (i.e., +Kaggle [6]), showing how this topic is sought after. There are already many data-sets to explore in this field, but no +clear winner algorithm that fits all scenarios with high accuracy, efficiency and simplicity (in terms of implementation +and interpretation). +A notable advancement in the field of NLP is the attention based transformers architecture [7]. This family of methods +excels in finding local connections between words, and better understanding the meaning of a sentence. A leading +example is the Bidirectional Encoder Representations from Transformers (BERT) [8] as well as its variations [9, 10, 11], +winning certain benchmarks [12, 13]. Several packages, such as Huggingface Transformers [14], make such models +accessible and easy to use as well as provide pre-trained versions. In addition, one can use transfer learning [15] to +further train BERT on their on data, creating a tailored model for the specific task at hand. +BERT, and often other transformer based models, are designed to handle text. They operate on the words of a given text +by encoding them into tokens, and by the connections between the tokens they learn the context of sentences. This +approach is limited, since sometimes more information can be extracted and used, not necessarily textual. Throughout +this paper we refer to this information as meta-data to distinguish it from the main stream of textual content (though one +may recognize it as the core data, depending on the application). For example, a meta-data could be the time stamp of +when the text was written, sent, published, etc. Another example is the writer of the text, when dealing with a small +arXiv:2301.01808v1 [cs.LG] 4 Jan 2023 + +Message classification using NLP and metadata +list of writers of a corpus. There have been some attempts to incorporate these into BERT models, for example by +assigning artificial tokens for writers or for temporal segments (token per month for example) [16]. This approach is +limited since not all meta-data entries are suitable for encoding by tokenization. In the example of temporal segments, +more segments introduce more tokens, leading to large computational resources consumption, and less segments cause +loss of information. Another approach is to concatenate the embeddings, created by the transformer module, with the +outputs of an embedding module for the meta-data. In this approach, a transformer for the text is trained (using direct or +transfer learning) on the text, and other separate modules (time series embedding, senders embeddings, etc.) are used +to embed the meta-data. All the embeddings are then concatenated and used as inputs to a classification network. A +drawback of this approach is that the internal network features are not trained from a combination of diffident input +streams, and therefore avoid cross dependent features (e.g. the importance of an email is not only determined by its +content, but also by who sent it, when, to whom else, attachments, etc.). +Novelty +To bridge these gaps, we implement a transformer based model that is able to train with both the text (transformer +architecture) and meta-data. We create a new architecture of a blocks based network. Each block handles different +kind of inputs. Splitting to blocks enables the flexibility to handle different kind of inputs. In addition, compared to +the standard practices that suggest separate training and implementation of a "voting between classifiers" method, the +proposed approach trains on the text and meta-data simultaneously, such that the language model (transformer) block +weights are adjusted based on the information passing through the meta-data classification block weights and vice versa. +We present results of the method with a main block based on a transformer that handles the text, and an additional block +that handles the pre-processed meta-data inputs individually. This method can be extended to support more complex +blocks, such as an advanced DL model for images [17], a temporal analysis block to extract information from temporal +meta-data [18], additional transformer blocks for multiple text inputs (for example, subject and body of an email), +categorical data, and more. To demonstrate the performance of the method we run multiple experiments on publicly +available data-sets (Amazon [19], Yelp [20], Reddit [21] and Enron [5]) to show the advantages of using the block +architecture, and compare them to the benchmarks in the literature (reviewed in the related work in section 2), which +are based on the transformer benchmark (BERT), Random Forest (RF) classifier, and Multi-Layer Perceptron (MLP) +networks. We achieve competitive results, and in most cases lead those benchmarks, showcasing that there is much to +extract from the meta-data compared to just using text for classification tasks. +2 +Related work +Natural language processing tasks. The publication of BERT [8] has been a turning point in the text classification +domain. The authors demonstrated high accuracy on complicated tasks such as question and answer, named entity +recognition, and textual entailment [13]. Since then, many authors investigated improved architectures and variations +such as RoBERTa [9], ALBERT [10], DistilBERT [11], and more. Some focus on better performance on the benchmark +tasks, and some create lighter versions of the model that reduce the computational demands while preserving competitive +accuracy. Other propositions, like XLNet [22] and GPT-3 [23], introduce competing architectures to BERT (also using +transformers). The benchmarks for these models are commonly GLUE, SuperGLUE [13], SQuAD 2.0 [12], and more. +Text classification is a less common benchmark, but the models can be used for this task as shown in this paper. +Accessibility of transformers. Another contributing factor to the growing popularity of transformers is the variety +of open-source code bases that make it easy for data-scientist to experiment with different architectures and then use +it in their applications. The Huggingface transformers package [24] is a Python library that can be used to train and +fine-tune models, with a large variety of base models to choose from, and straightforward implementation. The GPT-3 +[23] has been published as open source and, similar to several other implementations, offers a convenient application +programming interface (API). We mention that many libraries that do not use machine-learning for text classification +exist such as NLTK [25], spaCy [26], and more. These are also easily accessible and offer advanced NLP feature +extraction and other text analysis tools. +Text classification. There are many tasks in text classification, and each may be considered as a field of study. A +popular one is sentiment analysis, aiming to classify texts as positive or negative. The survey in [3] presents the +challenges in this domain and the latest innovations. Another example is the Spam or Ham task, where one tries to +differentiate a relevant email from irrelevant ones (like advertisements, phishing attempts, etc.) [1]. In this work we +investigate multi-label classification of messages. For example, classifying the category of a product based on purchase +review, the category of a thread based on the posts, the culinary specialty of a restaurant from customer reviews, the +type of product from purchase feedback, the category of an incoming email, and so on. For each of these tasks, publicly +2 + +Message classification using NLP and metadata +available data-sets exist and are used in this work to quantify the success of the proposed method. In addition, there are +many competitions in Kaggle [6] and other machine-learning research benchmark websites, using these data-sets. +Message classification with meta-data. There are two commonly used method to incorporate meta-data with textual +information for message categorization when using transformers. The first is concatenating the embeddings, computed +by the transformer, with the outputs of other embedding systems that are built specifically for the meta-data. In [27], the +authors use this approach for visual and audio meta-data. In [28] the authors use properties of the text as meta-data +and show that this approach can also work for the German language. In [29] the meta-data is the layout information of +scanned documents, and the authors propose an innovative architecture to extract information from both text and layout +information. There are many other studies exploring this approach. While it is simple to implement, this strategy has +several drawbacks. The training is usually done independently for the text and the meta-data, and the decisions are made +as a "voting between classifiers" approach. This may lead to conflicts, since the response of the text to the label may be +different from the response of the meta-data and the label, resulting in very low confidence predictions. We compare +the performance of the proposed method to this one in the results section. The second popular approach is to assign +tokens to the meta-data, and add this to the tokenized input of the transformer. In [16], in addition to a hierarchy of +labels, the authors introduce a method to inject multiple meta-data inputs with varying types (web, references, etc.) as +tokens to the embedding vector. Due to the simplicity of embedding the information using tokens, in terms of algorithm +and implementation, developers use this approach in their codes and it appears in many online notebooks and blogs +(open source codes) as well. The main drawback of these methods are the robustness to the input data. For example, +representing an image as a series of tokens is either done by encoding the image which usually faces loss of information, +or by utilizing a large number of tokens that exponentially highers the computational cost. In the numerical experiments +presented here, we do not compare to this method since the feature extraction we are using has a varying and potentially +high number of features, which lead to computational resource exhaustion when using this method. In this work, we +propose a method that can address the issues of the two popular methods, as described in the next section. +3 +Approach +We propose a method based on blocks to train a linguistic model with meta-data for a specific text classification task. +By splitting each type of meta-data input into different blocks, one can use state-of-the-art deep-learning architectures +to handle each meta-data type uniquely and more efficiently. In addition, the training is done using all block and in a +unified training loop, adjusting all the weights of all blocks in every optimizer step, so all information from the text and +meta-data sources contributes to the learning process. +3.1 +Blocks architecture +The transformer models, including BERT, can be used for text classification with the input text and corresponding +output labels. However, we claim that a lot of information can be found in the meta-data of the text. as can be seen in +Figure 1, we use the transformer model as a single block of a neural network. Then, we can add additional blocks for +dealing with the meta-data inputs. +With the recent developments in deep-learning, there are many advanced method of extracting information from input +signals for classification. For example, in [30] the authors discuss ways to use deep-learning for analysing time series +data. Messages typically have a temporal element, such as the time of arrival of an email, the time when a review +has been posted, a paper has been published, etc. We propose to utilize these advancements together for better model +training. +In Figure 1, an overall schematic view of the proposed approach is presented. In the first row, a standard transformer +architecture is illustrated [24]. The inputs are the tokenized messages, followed by the transformer layers that are +initially pre-trained. These layers are further trained (using transfer learning) to produce the embeddings, and a +classification layer is used to predict the categories of the messages. The transformer layers in this illustration are +presented using dashed lines to express the transfer learning process. The second row presents a meta-data extractor +using a deep-neural network. The green layers express layers trained for classification (for example, fully-connected +network architecture). The third row presents another transfer learning architecture. Each row expresses a different +block of a neural-network that handles different meta-data inputs. +3.1.1 +Representation +Using the propose approach, the expected representation of the data is much different than the one achieved by the +prominent approaches mentioned in section 2. Since the blocks are trained simultaneously, information from the +meta-data may impact the training of the core textual block (the transformer block), and vice-versa. Therefore, a new +3 + +Message classification using NLP and metadata +Figure 1: A sketch diagram of the block method. The first row illustrates a transformer architecture to handle textual +input, while the second and third rows illustrate neural networks for handling a specific meta-data input. All the modules +are then combines to create a unified prediction. Blue tiles are static and green tiles are trainable layers. The dashed +green tiles illustrate the transfer learning layers +and different representation is achieved. As a toy illustrative example, let us say that a single sample has text suggesting +a specific class but meta-data suggesting another, the textual block would train differently (and the textual representation +would be different, taking this in account), and the weights of the textual block would be able to capture this difference. +We illustrate the different in Figure 2. In Figure 2a we illustrate using tokens to inject the meta-data into the transformer +layers, producing an embedding (e1-eM1). In Figure2b we demonstrate concatenation of the embedding produced by +the transformer layers (e1-eM2) with a vector representation (v) of the meta-data. In Figure 2c the proposed blocks +approach is illustrated. The trainable blocks are wrapped in green to emphasize that training is done together, creating a +unified representation of the message, compared to 2b where the transformer layers and the dense layers are trained +independently, and the output representations are concatenated. We emphasize that the blocks may have different +architectures and may produce different vector representations, to produce a better representation. +3.1.2 +Combine +After the blocks, we propose to take a combination of the layers (e.g. averaging the outputs, summing the outputs, +concatenating the outputs, etc.). This combination may also be trained as part of the training loop. From the experimental +tests we find that the best performance was achieved by having the combine step as a weighted concatenation of the +outputs, producing an output sized as the number of classes times the number of blocks. We then use a few dense layers +to produce an output sized as the number of classes, and apply softmax activation for the classification. +4 +Experiments +For the numerical experiments we train the network of the proposed approach with two blocks: a) the transformer block +operating on the main text, and b) the meta-data block which is a fully-connected block operating on a one dimensional +meta-data vector. We emphasize that more advanced blocks can be used, but even this simple architecture provided +results that demonstrated the value from the meta-data and the blocks approach. In addition, it is simple to distinguish +4 + +Embeddings +Core text +block +matrix +Classification +result +Meta-data +block # 1 +Meta-data +Meta-data +block +block #n +embeddingsMessage classification using NLP and metadata +(a) Illustration of injecting meta-data as tokens +(b) Illustration of concatenation of meta-data +(c) Illustration of the blocks approach +Figure 2: Overview of the two common methods (a, b) and the proposed approach (c), emphasizing that the blocks +are trained together, so information from one block may affect the training of the other blocks (as opposed to (b) for +example, where the blocks are trained separately and concatenated) +between the transformer only architecture, and the transformer and meta-data architecture in this way, giving us better +explainability of the contribution of the meta-data. +4.1 +Data-sets +Amazon reviews. The Amazon product reviews data-set [19] contains a large number of product reviews, varying by +product category (label). The main text is the description of the review. We extract meta-data from the Amazon data-set +comes from the reviewer field (sender), the review creation date and time (timestamp), and the overall satisfaction +(enumerated). We expect correlations such as a user often reviews a specific category of products, or products more +frequently bought at a specific time of the day. These hold information about the product that is not available from the +text, and enhance the classification capabilities. This data-set has in total 82.83 million product reviews. We sub-sample +a subset of the reviews, by choosing the first 100,000 reviews of each category and then selecting the ones with the +longest texts. We eventually save 4,687 reviews for training, 521 for validation, and 2,064 for testing (eliminating +text-less messages). The final data-set is close to balanced (roughly the same number of reviews per category). +Yelp Open Data-set. The Yelp restaurant reviews data-set [20] contains restaurant reviews, varying by culinary class +(label). The main text is the restaurant review. We extract meta-data from the reviewer field (sender), the review +creation date and time (timestamp), is the review useful/funny/cool (enumerated), and the number of stars (enumerated). +The expected correlations here are reviewers that focus on specific restaurants, breakfast and dinner have different +timestamps, and properties of the review itself can suggest a rate for the experience. We clean the data-set similar to the +Amazon one, and save 4,612 reviews for training, 513 for validation, and 2,050 for testing, roughly balanced as well. +Reddit. The Reddit data-set [21] contains a large volume of Reddit posts, each belongs to a specific sub-Reddit (label). +Specifically, we use the data-set of version 2 from 2010. The main text is the post content. We extract meta-data +from the post uploader field (sender), the post upload date and time (timestamp), whether the post has comments +(enumerated), and whether it has attachments (enumerated). The expected correlations are that people post usually in +the same sub-Reddits, different poting times can be due to certain events in time, and the comment and attachment +properties often suggest the importance of the comment. We clean the data-set as well and save 4,950 posts for training, +550 for validation, and 2,200 for testing. We focus on 12 sub-Reddits (12 unique labels) and the data-set is roughly +balanced. +5 + +e1 +e2 +e3 +e4 +es +e6 +La +eg +eg +e10 +e11 +e12 +e13 +eM1 +Transformer layers +[CLS] my dog is cute [SEP] he likes play ##ing [SEP] [Sender] [Time]..e1 +e2 +e3 +e4 +es +e6 +e7 +eg +eM2 +V1 +V2 +VM3 +Transformer layers +Dense layers +[CLS] my dog is cute [SEP] he likes play ##ing [SEP] +Sender, Time..e1 +e2 +&a +e4 +es +e6 +e7 +eg +eg +e10 +eM4 +Transformer layers +Conv. layers +Dense layers +Dense layers +[CLS] my dog is cute [SEP] +Image data +Audio data +Block #n data +he likes play ##ing [SEP]Message classification using NLP and metadata +Enron Email Data-set. The Enron email data-set [5] contains a corpus of emails. Although this data-set is publicly +available, we had access to a limited internal version that has been tagged based on email category (label). The main +text is the email body. We extract meta-data from the email sender field (sender), and the email reception date and time +(timestamp). The expected correlations are that certain senders send emails in certain topics (colleagues send emails +about work, meetings, etc., while friends send emails about personal matters), and the time of arrival of the email is +usually correlated with the subject. We save 4,500 emails for training, 500 for validation, and 2,000 for testing. This +data-set is not balanced. +4.2 +Feature extractor +To conduct the numerical experiments we first discuss a genuine feature extractor we use to extract the meta-data +information from the available fields of the data-set. For each type of meta-data we extract corresponding features. +Since different data-sets have different fields, we use generic modules to extract information from different fields of the +same purpose, such as sender field (in emails data-set) or reviewer field (in restaurant reviews data-set). An overview of +the meta-data features we extract from the data-set is given in Table 1. Note, that the different data-sets have different +fields. Some of them do not have an enumerated field, and some have more than one. +Field type +Features +Number of slots +Description +Sender +Top senders +120 +One hot vector, +each slot represents if the sender +is one of the 100 to senders or not +Top affiliations +120 +Same with sender top affiliations +Sender frequency +1 +Compute the frequency of +messages received from this sender +Timestamp +Day +7 +One hot vector, each slot +represents if the message arrived +at that day +Working hours +1 +Indicating if the message +arrived within working hours +or not +Rush hour +50 +Creating a histogram of 50 bins +and a corresponding 50 slots +one hot vector where each slot +represents if the message arrived +in the corresponding bin time +Enumerated +Enumerated value +#Options +One hot vector with 1 +in the slot representing +the option and 0 otherwise +Numeric +Numeric value +1 +The numeric value +Table 1: Review of meta-data features extraction per field +4.3 +Training details +We use a simple network architecture so that we can clearly show the advantages of the method. We emphasize that +with more advanced handling of the meta-data and introduction of more blocks, higher accuracy is expected. The +model used for the text embedding is a pre-trained BERT model from the HuggingFace [14] library (bert-base-uncased). +We use transfer learning to further train the language model with the meta-data, given the new task (new data). In +addition, we used the ktrain package, specifically the text classification modules, and re-wrote them to have a message +classification class. For the meta-data classification block we use two fully-connected layers, the first is of the size +of the input meta-data and the size of the second is the number of classes, both use the Rectified Linear Unit (ReLU) +activation. After the blocks we either use a Random Forest classifier (with 250 trees, unconstrained depth, up to one +sample per leaf, and splitting criterion based on the Gini index) or combine the outputs into one (with averaging or +using trainable weights), depending on the test scenario. In the case of output weighting, we use one fully connected +layer at the end, its size is the number of classes and it is followed by a softmax activation (for the classification). +4.4 +Results +We use the data-sets mentioned in Section 4.1. We compare multiple methods in terms of accuracy: +6 + +Message classification using NLP and metadata +1. Pre-trained BERT with an additional fully-connected layer. +2. Pre-trained BERT with a random forest classifier. +3. Pre-trained BERT with concatenated meta-data and an additional fully-connected layer. +4. Pre-trained BERT with concatenated meta-data and a random forest classifier. +5. Transfer learning BERT with an additional fully-connected layer. +6. Transfer learning BERT with a random forest classifier. +7. Transfer learning BERT with concatenated meta-data and an additional fully-connected layer. +8. Transfer learning BERT with concatenated meta-data and a random forest classifier. +9. The proposed method (BERT and meta-data) with output averaging. +10. The proposed method (BERT and meta-data) with output weighting. +Methods 1-4 use embeddings from a pre-trained BERT model (without transfer learning), and train either an extra +fully-connected layer or a random forest classifier [31] for the classification. We do this either without (1-2) or with +(3-4) concatenation of the meta-data to the embeddings, following the common methods in the literature. Methods +5-8 are similar, but the BERT transformers layers are further trained on the specific task data. Methods 3,4,7,8 are +the reference methods of concatenating the BERT model embedding with a meta-data embedding from the literature +[16]. Methods 9-10 are the proposed ones, exploring the effect of combining the transformer output with the meta-data +block output, once with averaging the block outputs and once with a fully-connected layer to learn the weights after +concatenating the output representations of each block. The latter involves training to learn this weight, which is +implemented so that it happens in the same training process as all other weights in the network. The results are given in +Table 2. By observing the results, we see that the proposed method is competitive with all other methods and even +outperforms most of them. +Method# +Amazon reviews +Yelp Open Data-set +Reddit +Enron Emails +1 +0.66 +0.29 +0.56 +0.49 +2 +0.61 +0.22 +0.52 +0.49 +3 +0.65 +0.24 +0.46 +0.48 +4 +0.61 +0.22 +0.5 +0.5 +5 +0.74 +0.39 +0.61 +0.47 +6 +0.73 +0.38 +0.6 +0.47 +7 +0.71 +0.38 +0.62 +0.47 +8 +0.73 +0.39 +0.6 +0.47 +9 +0.7 +0.3 +0.62 +0.47 +10 +0.77 +0.4 +0.62 +0.53 +Table 2: Results table +An interesting result is observed for the Enron emails data-set. We notice that the pre-trained methods performed better +than the transfer-learning based methods. This can be explained by the mismatches between the textual information +and the meta-data information. While the text suggests one class, the meta-data may suggest a different one. Methods +1-8, in this case, act as voting between classifiers. However, as mentioned in section 3.1.1, the proposed method is +influenced by both text and meta-data and learns a better representation that can take into account these differences, +performing better than all other methods. +5 +Conclusion +In this paper we proposed a new framework for training classification models. The proposition relies on the availability +of additional, not necessarily textual, data channels such as attached images, audio, sender and timestamp, etc. We +proposed an architecture with which one can utilize the aforementioned additional information using different blocks, +along with the text, and train a neural network to perform message classification more accurately. We demonstrate +the strength of this method using a set examples, varying by data-set and classification algorithm, and show that the +proposed method outperforms the reference related works. +7 + +Message classification using NLP and metadata +References +[1] Asif Karim, Sami Azam, Bharanidharan Shanmugam, and Krishnan Kannoorpatti. Efficient clustering of emails +into spam and ham: The foundational study of a comprehensive unsupervised framework. IEEE Access, 8:154759– +154788, 2020. +[2] Jianglei Han, Jing Li, and Aixin Sun. Uftr: A unified framework for ticket routing, 2020. +[3] Walaa Medhat, Ahmed Hassan, and Hoda Korashy. Sentiment analysis algorithms and applications: A survey. +Ain Shams engineering journal, 5(4):1093–1113, 2014. +[4] Dheeraj Gupta, P.S. Joshi, A.K. Bhattacharjee, and R.S. Mundada. Ids alerts classification using knowledge-based +evaluation. 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Random forests. Machine learning, 45(1):5–32, 2001. +9 + diff --git a/YNAzT4oBgHgl3EQf1_6A/content/tmp_files/load_file.txt b/YNAzT4oBgHgl3EQf1_6A/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..107a09ae16dc9236adeba4dcf6786bf1a939f1a6 --- /dev/null +++ b/YNAzT4oBgHgl3EQf1_6A/content/tmp_files/load_file.txt @@ -0,0 +1,434 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf,len=433 +page_content='MESSAGENET: MESSAGE CLASSIFICATION USING NATURAL LANGUAGE PROCESSING AND META-DATA Adar Kahana, Oren Elisha Microsoft R&D center, ILDC {adarkahana, orelisha}@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='com ABSTRACT In this paper we propose a new Deep Learning (DL) approach for message classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Our method is based on the state-of-the-art Natural Language Processing (NLP) building blocks, combined with a novel technique for infusing the meta-data input that is typically available in messages such as the sender information, timestamps, attached image, audio, affiliations, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' As we demonstrate throughout the paper, going beyond the mere text by leveraging all available channels in the message, could yield an improved representation and higher classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' To achieve message representation, each type of input is processed in a dedicated block in the neural network architecture that is suitable for the data type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Such an implementation enables training all blocks together simultaneously, and forming cross channels features in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We show in the Experiments Section that in some cases, message’s meta-data holds an additional information that cannot be extracted just from the text, and when using this information we achieve better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Furthermore, we demonstrate that our multi-modality block approach outperforms other approaches for injecting the meta data to the the text classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Keywords Message classification · Meta data injection · Deep learning · Natural language processing 1 Introduction Many real world applications require message classification and regression, such as handling spam emails [1], ticket routing [2], article sentiment review [3] and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Accurate message classification could improve critical scenarios such as in call centers (routing tickets based on topic) [2], alert systems (flagging highly important alert messages) [4], and categorizing incoming messages (automatically unclutter emails) [1, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The main distinction between text and message classification is the availability of additional attributes, such as the sender information, timestamps, attached image, audio, affiliations, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' New message classification contests often appear in the prominent platforms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=', Kaggle [6]), showing how this topic is sought after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' There are already many data-sets to explore in this field, but no clear winner algorithm that fits all scenarios with high accuracy, efficiency and simplicity (in terms of implementation and interpretation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' A notable advancement in the field of NLP is the attention based transformers architecture [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' This family of methods excels in finding local connections between words, and better understanding the meaning of a sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' A leading example is the Bidirectional Encoder Representations from Transformers (BERT) [8] as well as its variations [9, 10, 11], winning certain benchmarks [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Several packages, such as Huggingface Transformers [14], make such models accessible and easy to use as well as provide pre-trained versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In addition, one can use transfer learning [15] to further train BERT on their on data, creating a tailored model for the specific task at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' BERT, and often other transformer based models, are designed to handle text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' They operate on the words of a given text by encoding them into tokens, and by the connections between the tokens they learn the context of sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' This approach is limited, since sometimes more information can be extracted and used, not necessarily textual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Throughout this paper we refer to this information as meta-data to distinguish it from the main stream of textual content (though one may recognize it as the core data, depending on the application).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' For example, a meta-data could be the time stamp of when the text was written, sent, published, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Another example is the writer of the text, when dealing with a small arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='01808v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='LG] 4 Jan 2023 Message classification using NLP and metadata list of writers of a corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' There have been some attempts to incorporate these into BERT models, for example by assigning artificial tokens for writers or for temporal segments (token per month for example) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' This approach is limited since not all meta-data entries are suitable for encoding by tokenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In the example of temporal segments, more segments introduce more tokens, leading to large computational resources consumption, and less segments cause loss of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Another approach is to concatenate the embeddings, created by the transformer module, with the outputs of an embedding module for the meta-data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In this approach, a transformer for the text is trained (using direct or transfer learning) on the text, and other separate modules (time series embedding, senders embeddings, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=') are used to embed the meta-data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' All the embeddings are then concatenated and used as inputs to a classification network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' A drawback of this approach is that the internal network features are not trained from a combination of diffident input streams, and therefore avoid cross dependent features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' the importance of an email is not only determined by its content, but also by who sent it, when, to whom else, attachments, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Novelty To bridge these gaps, we implement a transformer based model that is able to train with both the text (transformer architecture) and meta-data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We create a new architecture of a blocks based network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Each block handles different kind of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Splitting to blocks enables the flexibility to handle different kind of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In addition, compared to the standard practices that suggest separate training and implementation of a "voting between classifiers" method, the proposed approach trains on the text and meta-data simultaneously, such that the language model (transformer) block weights are adjusted based on the information passing through the meta-data classification block weights and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We present results of the method with a main block based on a transformer that handles the text, and an additional block that handles the pre-processed meta-data inputs individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' This method can be extended to support more complex blocks, such as an advanced DL model for images [17], a temporal analysis block to extract information from temporal meta-data [18], additional transformer blocks for multiple text inputs (for example, subject and body of an email), categorical data, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' To demonstrate the performance of the method we run multiple experiments on publicly available data-sets (Amazon [19], Yelp [20], Reddit [21] and Enron [5]) to show the advantages of using the block architecture, and compare them to the benchmarks in the literature (reviewed in the related work in section 2), which are based on the transformer benchmark (BERT), Random Forest (RF) classifier, and Multi-Layer Perceptron (MLP) networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We achieve competitive results, and in most cases lead those benchmarks, showcasing that there is much to extract from the meta-data compared to just using text for classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 2 Related work Natural language processing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The publication of BERT [8] has been a turning point in the text classification domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The authors demonstrated high accuracy on complicated tasks such as question and answer, named entity recognition, and textual entailment [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Since then, many authors investigated improved architectures and variations such as RoBERTa [9], ALBERT [10], DistilBERT [11], and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Some focus on better performance on the benchmark tasks, and some create lighter versions of the model that reduce the computational demands while preserving competitive accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Other propositions, like XLNet [22] and GPT-3 [23], introduce competing architectures to BERT (also using transformers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The benchmarks for these models are commonly GLUE, SuperGLUE [13], SQuAD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='0 [12], and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Text classification is a less common benchmark, but the models can be used for this task as shown in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Accessibility of transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Another contributing factor to the growing popularity of transformers is the variety of open-source code bases that make it easy for data-scientist to experiment with different architectures and then use it in their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The Huggingface transformers package [24] is a Python library that can be used to train and fine-tune models, with a large variety of base models to choose from, and straightforward implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The GPT-3 [23] has been published as open source and, similar to several other implementations, offers a convenient application programming interface (API).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We mention that many libraries that do not use machine-learning for text classification exist such as NLTK [25], spaCy [26], and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' These are also easily accessible and offer advanced NLP feature extraction and other text analysis tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' There are many tasks in text classification, and each may be considered as a field of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' A popular one is sentiment analysis, aiming to classify texts as positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The survey in [3] presents the challenges in this domain and the latest innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Another example is the Spam or Ham task, where one tries to differentiate a relevant email from irrelevant ones (like advertisements, phishing attempts, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=') [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In this work we investigate multi-label classification of messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' For example, classifying the category of a product based on purchase review, the category of a thread based on the posts, the culinary specialty of a restaurant from customer reviews, the type of product from purchase feedback, the category of an incoming email, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' For each of these tasks, publicly 2 Message classification using NLP and metadata available data-sets exist and are used in this work to quantify the success of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In addition, there are many competitions in Kaggle [6] and other machine-learning research benchmark websites, using these data-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Message classification with meta-data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' There are two commonly used method to incorporate meta-data with textual information for message categorization when using transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The first is concatenating the embeddings, computed by the transformer, with the outputs of other embedding systems that are built specifically for the meta-data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In [27], the authors use this approach for visual and audio meta-data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In [28] the authors use properties of the text as meta-data and show that this approach can also work for the German language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In [29] the meta-data is the layout information of scanned documents, and the authors propose an innovative architecture to extract information from both text and layout information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' There are many other studies exploring this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' While it is simple to implement, this strategy has several drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The training is usually done independently for the text and the meta-data, and the decisions are made as a "voting between classifiers" approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' This may lead to conflicts, since the response of the text to the label may be different from the response of the meta-data and the label, resulting in very low confidence predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We compare the performance of the proposed method to this one in the results section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The second popular approach is to assign tokens to the meta-data, and add this to the tokenized input of the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In [16], in addition to a hierarchy of labels, the authors introduce a method to inject multiple meta-data inputs with varying types (web, references, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=') as tokens to the embedding vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Due to the simplicity of embedding the information using tokens, in terms of algorithm and implementation, developers use this approach in their codes and it appears in many online notebooks and blogs (open source codes) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The main drawback of these methods are the robustness to the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' For example, representing an image as a series of tokens is either done by encoding the image which usually faces loss of information, or by utilizing a large number of tokens that exponentially highers the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In the numerical experiments presented here, we do not compare to this method since the feature extraction we are using has a varying and potentially high number of features, which lead to computational resource exhaustion when using this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In this work, we propose a method that can address the issues of the two popular methods, as described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 3 Approach We propose a method based on blocks to train a linguistic model with meta-data for a specific text classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' By splitting each type of meta-data input into different blocks, one can use state-of-the-art deep-learning architectures to handle each meta-data type uniquely and more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In addition, the training is done using all block and in a unified training loop, adjusting all the weights of all blocks in every optimizer step, so all information from the text and meta-data sources contributes to the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='1 Blocks architecture The transformer models, including BERT, can be used for text classification with the input text and corresponding output labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' However, we claim that a lot of information can be found in the meta-data of the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' as can be seen in Figure 1, we use the transformer model as a single block of a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Then, we can add additional blocks for dealing with the meta-data inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' With the recent developments in deep-learning, there are many advanced method of extracting information from input signals for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' For example, in [30] the authors discuss ways to use deep-learning for analysing time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Messages typically have a temporal element, such as the time of arrival of an email, the time when a review has been posted, a paper has been published, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We propose to utilize these advancements together for better model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In Figure 1, an overall schematic view of the proposed approach is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In the first row, a standard transformer architecture is illustrated [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The inputs are the tokenized messages, followed by the transformer layers that are initially pre-trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' These layers are further trained (using transfer learning) to produce the embeddings, and a classification layer is used to predict the categories of the messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The transformer layers in this illustration are presented using dashed lines to express the transfer learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The second row presents a meta-data extractor using a deep-neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The green layers express layers trained for classification (for example, fully-connected network architecture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The third row presents another transfer learning architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Each row expresses a different block of a neural-network that handles different meta-data inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='1 Representation Using the propose approach, the expected representation of the data is much different than the one achieved by the prominent approaches mentioned in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Since the blocks are trained simultaneously, information from the meta-data may impact the training of the core textual block (the transformer block), and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Therefore, a new 3 Message classification using NLP and metadata Figure 1: A sketch diagram of the block method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The first row illustrates a transformer architecture to handle textual input, while the second and third rows illustrate neural networks for handling a specific meta-data input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' All the modules are then combines to create a unified prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Blue tiles are static and green tiles are trainable layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The dashed green tiles illustrate the transfer learning layers and different representation is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' As a toy illustrative example, let us say that a single sample has text suggesting a specific class but meta-data suggesting another, the textual block would train differently (and the textual representation would be different, taking this in account), and the weights of the textual block would be able to capture this difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We illustrate the different in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In Figure 2a we illustrate using tokens to inject the meta-data into the transformer layers, producing an embedding (e1-eM1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In Figure2b we demonstrate concatenation of the embedding produced by the transformer layers (e1-eM2) with a vector representation (v) of the meta-data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In Figure 2c the proposed blocks approach is illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The trainable blocks are wrapped in green to emphasize that training is done together, creating a unified representation of the message, compared to 2b where the transformer layers and the dense layers are trained independently, and the output representations are concatenated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We emphasize that the blocks may have different architectures and may produce different vector representations, to produce a better representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='2 Combine After the blocks, we propose to take a combination of the layers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' averaging the outputs, summing the outputs, concatenating the outputs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' This combination may also be trained as part of the training loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' From the experimental tests we find that the best performance was achieved by having the combine step as a weighted concatenation of the outputs, producing an output sized as the number of classes times the number of blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We then use a few dense layers to produce an output sized as the number of classes, and apply softmax activation for the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 4 Experiments For the numerical experiments we train the network of the proposed approach with two blocks: a) the transformer block operating on the main text, and b) the meta-data block which is a fully-connected block operating on a one dimensional meta-data vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We emphasize that more advanced blocks can be used, but even this simple architecture provided results that demonstrated the value from the meta-data and the blocks approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' it is simple to distinguish 4 Embeddings Core text block matrix Classification result Meta-data block # 1 Meta-data Meta-data block block #n embeddingsMessage classification using NLP and metadata (a) Illustration of injecting meta-data as tokens (b) Illustration of concatenation of meta-data (c) Illustration of the blocks approach Figure 2: Overview of the two common methods (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' b) and the proposed approach (c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' emphasizing that the blocks are trained together,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' so information from one block may affect the training of the other blocks (as opposed to (b) for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' where the blocks are trained separately and concatenated) between the transformer only architecture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' and the transformer and meta-data architecture in this way,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' giving us better explainability of the contribution of the meta-data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='1 Data-sets Amazon reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The Amazon product reviews data-set [19] contains a large number of product reviews, varying by product category (label).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The main text is the description of the review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We extract meta-data from the Amazon data-set comes from the reviewer field (sender), the review creation date and time (timestamp), and the overall satisfaction (enumerated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We expect correlations such as a user often reviews a specific category of products, or products more frequently bought at a specific time of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' These hold information about the product that is not available from the text, and enhance the classification capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' This data-set has in total 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='83 million product reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We sub-sample a subset of the reviews, by choosing the first 100,000 reviews of each category and then selecting the ones with the longest texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We eventually save 4,687 reviews for training, 521 for validation, and 2,064 for testing (eliminating text-less messages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The final data-set is close to balanced (roughly the same number of reviews per category).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Yelp Open Data-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The Yelp restaurant reviews data-set [20] contains restaurant reviews, varying by culinary class (label).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The main text is the restaurant review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We extract meta-data from the reviewer field (sender), the review creation date and time (timestamp), is the review useful/funny/cool (enumerated), and the number of stars (enumerated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The expected correlations here are reviewers that focus on specific restaurants, breakfast and dinner have different timestamps, and properties of the review itself can suggest a rate for the experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We clean the data-set similar to the Amazon one, and save 4,612 reviews for training, 513 for validation, and 2,050 for testing, roughly balanced as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Reddit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The Reddit data-set [21] contains a large volume of Reddit posts, each belongs to a specific sub-Reddit (label).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Specifically, we use the data-set of version 2 from 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The main text is the post content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We extract meta-data from the post uploader field (sender), the post upload date and time (timestamp), whether the post has comments (enumerated), and whether it has attachments (enumerated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The expected correlations are that people post usually in the same sub-Reddits, different poting times can be due to certain events in time, and the comment and attachment properties often suggest the importance of the comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We clean the data-set as well and save 4,950 posts for training, 550 for validation, and 2,200 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We focus on 12 sub-Reddits (12 unique labels) and the data-set is roughly balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 5 e1 e2 e3 e4 es e6 La eg eg e10 e11 e12 e13 eM1 Transformer layers [CLS] my dog is cute [SEP] he likes play ##ing [SEP] [Sender] [Time].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='.e1 e2 e3 e4 es e6 e7 eg eM2 V1 V2 VM3 Transformer layers Dense layers [CLS] my dog is cute [SEP] he likes play ##ing [SEP] Sender, Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='.e1 e2 &a e4 es e6 e7 eg eg e10 eM4 Transformer layers Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' layers Dense layers Dense layers [CLS] my dog is cute [SEP] Image data Audio data Block #n data he likes play ##ing [SEP]Message classification using NLP and metadata Enron Email Data-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The Enron email data-set [5] contains a corpus of emails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Although this data-set is publicly available, we had access to a limited internal version that has been tagged based on email category (label).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The main text is the email body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We extract meta-data from the email sender field (sender), and the email reception date and time (timestamp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The expected correlations are that certain senders send emails in certain topics (colleagues send emails about work, meetings, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=', while friends send emails about personal matters), and the time of arrival of the email is usually correlated with the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We save 4,500 emails for training, 500 for validation, and 2,000 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' This data-set is not balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='2 Feature extractor To conduct the numerical experiments we first discuss a genuine feature extractor we use to extract the meta-data information from the available fields of the data-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' For each type of meta-data we extract corresponding features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Since different data-sets have different fields, we use generic modules to extract information from different fields of the same purpose, such as sender field (in emails data-set) or reviewer field (in restaurant reviews data-set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' An overview of the meta-data features we extract from the data-set is given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Note, that the different data-sets have different fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Some of them do not have an enumerated field, and some have more than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Field type Features Number of slots Description Sender Top senders 120 One hot vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' each slot represents if the sender is one of the 100 to senders or not Top affiliations 120 Same with sender top affiliations Sender frequency 1 Compute the frequency of messages received from this sender Timestamp Day 7 One hot vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' each slot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='represents if the message arrived ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='at that day ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='Working hours ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='Indicating if the message ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='arrived within working hours ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='or not ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='Rush hour ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='Creating a histogram of 50 bins ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='and a corresponding 50 slots ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='one hot vector where each slot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='represents if the message arrived ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='in the corresponding bin time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='Enumerated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='Enumerated value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='#Options ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='One hot vector with 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='in the slot representing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='the option and 0 otherwise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='Numeric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='Numeric value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='The numeric value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='Table 1: Review of meta-data features extraction per field ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='3 Training details We use a simple network architecture so that we can clearly show the advantages of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We emphasize that with more advanced handling of the meta-data and introduction of more blocks, higher accuracy is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The model used for the text embedding is a pre-trained BERT model from the HuggingFace [14] library (bert-base-uncased).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We use transfer learning to further train the language model with the meta-data, given the new task (new data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In addition, we used the ktrain package, specifically the text classification modules, and re-wrote them to have a message classification class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' For the meta-data classification block we use two fully-connected layers, the first is of the size of the input meta-data and the size of the second is the number of classes, both use the Rectified Linear Unit (ReLU) activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' After the blocks we either use a Random Forest classifier (with 250 trees, unconstrained depth, up to one sample per leaf, and splitting criterion based on the Gini index) or combine the outputs into one (with averaging or using trainable weights), depending on the test scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In the case of output weighting, we use one fully connected layer at the end, its size is the number of classes and it is followed by a softmax activation (for the classification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='4 Results We use the data-sets mentioned in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We compare multiple methods in terms of accuracy: 6 Message classification using NLP and metadata 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Pre-trained BERT with an additional fully-connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Pre-trained BERT with a random forest classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Pre-trained BERT with concatenated meta-data and an additional fully-connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Pre-trained BERT with concatenated meta-data and a random forest classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Transfer learning BERT with an additional fully-connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Transfer learning BERT with a random forest classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Transfer learning BERT with concatenated meta-data and an additional fully-connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Transfer learning BERT with concatenated meta-data and a random forest classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The proposed method (BERT and meta-data) with output averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The proposed method (BERT and meta-data) with output weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Methods 1-4 use embeddings from a pre-trained BERT model (without transfer learning), and train either an extra fully-connected layer or a random forest classifier [31] for the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We do this either without (1-2) or with (3-4) concatenation of the meta-data to the embeddings, following the common methods in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Methods 5-8 are similar, but the BERT transformers layers are further trained on the specific task data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Methods 3,4,7,8 are the reference methods of concatenating the BERT model embedding with a meta-data embedding from the literature [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Methods 9-10 are the proposed ones, exploring the effect of combining the transformer output with the meta-data block output, once with averaging the block outputs and once with a fully-connected layer to learn the weights after concatenating the output representations of each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The latter involves training to learn this weight, which is implemented so that it happens in the same training process as all other weights in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The results are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' By observing the results, we see that the proposed method is competitive with all other methods and even outperforms most of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Method# Amazon reviews Yelp Open Data-set Reddit Enron Emails 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='49 2 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='53 Table 2: Results table An interesting result is observed for the Enron emails data-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We notice that the pre-trained methods performed better than the transfer-learning based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' This can be explained by the mismatches between the textual information and the meta-data information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' While the text suggests one class, the meta-data may suggest a different one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Methods 1-8, in this case, act as voting between classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' However, as mentioned in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content='1, the proposed method is influenced by both text and meta-data and learns a better representation that can take into account these differences, performing better than all other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 5 Conclusion In this paper we proposed a new framework for training classification models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' The proposition relies on the availability of additional, not necessarily textual, data channels such as attached images, audio, sender and timestamp, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We proposed an architecture with which one can utilize the aforementioned additional information using different blocks, along with the text, and train a neural network to perform message classification more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' We demonstrate the strength of this method using a set examples, varying by data-set and classification algorithm, and show that the proposed method outperforms the reference related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' 7 Message classification using NLP and metadata References [1] Asif Karim, Sami Azam, Bharanidharan Shanmugam, and Krishnan Kannoorpatti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Efficient clustering of emails into spam and ham: The foundational study of a comprehensive unsupervised framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' IEEE Access, 8:154759– 154788, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' [2] Jianglei Han, Jing Li, and Aixin Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} 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Matthew Honnibal and Ines Montani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' spaCy 2: Natural language understanding with Bloom embeddings, convolutional neural networks and incremental parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' To appear, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' [27] Tanzila Rahman, Mengyu Yang, and Leonid Sigal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Tribert: Full-body human-centric audio-visual representation learning for visual sound separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' arXiv 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' [29] Yiheng Xu, Minghao Li, Lei Cui, Shaohan Huang, Furu Wei, and Ming Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Layoutlm: Pre-training of text and layout for document image understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pages 1192–1200, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' [30] John-Syin Ang, Kok-Why Ng, and Fang-Fang Chua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAzT4oBgHgl3EQf1_6A/content/2301.01808v1.pdf'} +page_content=' Modeling time series 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a/a9E0T4oBgHgl3EQfngEk/content/tmp_files/2301.02512v1.pdf.txt b/a9E0T4oBgHgl3EQfngEk/content/tmp_files/2301.02512v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..990b554a25e2111a82f8dc25b5ad27b97870c780 --- /dev/null +++ b/a9E0T4oBgHgl3EQfngEk/content/tmp_files/2301.02512v1.pdf.txt @@ -0,0 +1,1551 @@ +D-Algebraic Functions +Rida Ait El Manssour, Anna-Laura Sattelberger, Bertrand Teguia Tabuguia +Abstract +Differentially-algebraic (D-algebraic) functions are solutions of polynomial equa- +tions in the function, its derivatives, and the independent variables. We revisit closure +properties of these functions by providing constructive proofs. We present algorithms +to compute algebraic differential equations for compositions and arithmetic manipula- +tions of univariate D-algebraic functions and derive bounds for the order of the resulting +differential equations. We apply our methods to examples in the sciences. +1 +Introduction +The Weyl algebra encodes linear differential operators with polynomial coefficients, such as +the operator P = ∂2 − x arising from Airy’s differential equation +f ′′(x) − xf(x) = 0 . +(1.1) +Solutions of such differential equations are called holonomic functions. Differential algebra +investigates polynomials in a differential indeterminate y and its derivatives with coefficients +in a differential ring. Those polynomials are called differential polynomials and their as- +sociated differential equations are commonly referred to as algebraic differential equations +(ADEs). For instance, the differential polynomial p = y′2 − 4y3 − c1y − c2, where c1, c2 are +constants, encodes the differential equation +(y′(x))2 = 4(y(x))3 + c1y(x) + c2 , +(1.2) +of which the Weierstrass elliptic function ℘ (see [24, 13]) is a solution. +ADEs also nat- +urally arise in structural identifiability, cf. [5]. +Functions which are zeros of differential +polynomials—or, equivalently, solutions of the corresponding ADEs—are called D-algebraic +functions. This class of functions arises in a natural way from the study of holonomic func- +tions: for instance, the reciprocal of a holonomic function is in general not holonomic, but +it is D-algebraic. The present article can hence be located at the interim of the theory of +D-ideals and differential algebra. For introductions to those fields, we refer our readers to +[29, 28] and [25, 14], respectively. Although they are present at several places in the literature +(see [26, 31, 27, 15]), it is difficult to find exhaustive expositions about D-algebraic functions. +This could be explained by the challenging computational complexity observable from [11], +which is detrimental for applications. One attempt of computer algebra is to find subclasses +1 +arXiv:2301.02512v1 [math.AG] 6 Jan 2023 + +of D-algebraic functions that offer both a mathematical structure and efficient algorithms +for the corresponding arithmetic. Some relevant subclasses are holonomic functions, which +are also called “D-finite functions” (see [17, 32]), DD-finite functions that satisfy differential +equations with D-finite coefficients [12], or functions that satisfy ADEs of degree at most 2 +[34, 33], called “δ2-finite functions” therein. While the rich theory of D-modules and holo- +nomic functions covers the degree-one case, similar constructions for ADEs of higher degree +are still in an early stage. The idea of bounding the degree of the ADEs is to alleviate the +complexity compared to the general case. The class of D-algebraic functions has nice closure +properties, which is for instance carried out in [23] and [35]. The former illustrates the use of +D-algebraic functions in combinatorics for studying generating functions, with an emphasis +on quadrant walks via Tutte’s invariant method. The latter focuses on the zero test prob- +lem for D-algebraic functions viewed as formal power series. The earliest appearance of the +terminology “differentially algebraic” that we are aware of is in Rubel’s work [27]. +In this article, we construct ADEs for compositions and rational functions of D-algebraic +functions, taking only the ADEs of the original functions as input. We give bounds for +the order of the resulting differential polynomials. Finding an ADE fulfilled by a rational +expression in a D-algebraic function f, or finding an ADE fulfilled by its antiderivative, are +unary operations in the input ADE. The rest of operations—like sums, products, ratios, +and compositions—are binary operations. However, we implemented them for arbitrarily +many operands. Our algorithms take differential polynomials p, q as input. From those, we +construct ADEs which rational functions, antiderivatives, and compositions of all solutions +of the input ADEs fulfill. We stress that working with the differential polynomials implies +working with the set of all solutions of the corresponding ADEs. +Gr¨obner basis theories exist both for D-ideals and for differential ideals. Viewing ADEs +as differential polynomials, one can encode an operation between D-algebraic functions by an +ideal in a differential polynomial ring. Such an ideal is differentially generated by the given +polynomials and a rational expression built from the underlying operation. We use jets to +truncate the obtained differential ideal at desired order before applying elimination theory +based on Gr¨obner basis techniques. We compare it to a second method that still proceeds +with Gr¨obner bases, but uses bounds for the order of the resulting differential polynomial, +which we establish in our study. The main contribution of this article is the development of +two strategies to compute differential equations satisfied by rational expressions and com- +positions of D-algebraic functions. In particular, our algorithms and their implementations +are general, reliable, and outperform existing algorithms and software, which often do not +contain such computations. For instance, the find_ioequations command of the Julia [2] +package StructuralIdentifiability can be used to derive ADEs with constant coeffi- +cients only; moreover, that command requires a user-defined dynamical model, which is not +needed in our algorithms for arithmetic operations and compositions. For each operation +with D-algebraic functions, we provide a bound for the order of the resulting differential +polynomial in Theorems 4.9 and 4.12 which depends on those of the input ADEs. We imple- +mented our first method in Macaulay2 [9]. The second one is implemented as the Maple [20] +package NLDE. Our implementations are made available via the MathRepo [6]—a repository +website hosted by MPI MiS—at https://mathrepo.mis.mpg.de/DAlgebraicFunctions. +2 + +Outline. Our article is organized as follows. Section 2 recalls basic concepts about the +Weyl algebra as well as differential algebra. In Section 3, we present D-algebraic functions +and investigate their closure properties. +In Section 4, we study the arithmetic of those +functions: we construct ADEs for antiderivatives, compositions and rational expressions of +D-algebraic functions and study the order of the resulting differential equations. Section 5 +presents pseudocode to carry those operations out in practice. We present applications of +our results in the study of Feynman integrals and epidemiology. The code for our implemen- +tations in Macaulay2 and Maple is provided in Appendices A and B, respectively. +Notation. By N, we denote the nonnegative integers. Letters f, g, h are reserved for +functions, D denotes the Weyl algebra, P ∈ D is used for linear differential operators. Dif- +ferential polynomials are denoted by lower case letters p, q, r, s. They are polynomials in +dependent variables u, y, z, w and the independent variable x (or x1, . . . , xk in the multivari- +ate case). “Degree” means the total degree of polynomials, unless stated otherwise. By a +rational expression in f, we will mean a rational function in x, f, and the derivatives of f. +2 +Algebraic aspects of differential equations +We briefly recall some algebraic aspects of differential equations. We here revisit two classes +of differential equations, namely linear and algebraic differential equations with polynomial +coefficients. Throughout this section, we describe the case of univariate functions. +2.1 +The Weyl algebra and holonomic functions +The Weyl algebra, denoted D := K[x]⟨∂⟩, is the free K-algebra generated by x and ∂ modulo +the following relation: the commutator of ∂ and x is +[∂, x] = ∂x − x∂ = 1 , +(2.1) +which encodes Leibniz’ rule for taking the derivative of a product of functions in a formal way. +Hence +D = +� k +� +i=0 +ai∂i | k ∈ N, ai ∈ K[x] +� +. +(2.2) +The order of a differential operator P = � +i ai∂i is +ord(P) := max {i | ai ̸= 0} . +(2.3) +For P ∈ D, the vanishing locus of the leading polynomial aord(P) is the singular locus of P. +To a differential operator P, one associates the ODE P(f) = 0, i.e., one looks for functions f +that are annihilated by the differential operator P. +Definition 2.1. A function f(x) is holonomic if there exists a differential operator P ∈ D +which annihilates f. +3 + +Holonomicity of D-modules dates back to Bernstein and Kashiwara. Effective compu- +tations with holonomic functions were first studied by Zeilberger [36] for proving identities +between special functions automatically. The name “D-finite” is justified by the following +observation: a function f is holonomic if and only if the K(x)-vector space spanned by the +derivatives of f is finite-dimensional, i.e., if +dimK(x) +� +spanK(x) +�� +∂k(f) +� +k∈N +�� +< ∞ . +(2.4) +In general, a (uni- or multivariate) function is called holonomic if its annihilating D-ideal +is holonomic. +We refer to [28, 29] for an introduction to D-modules with a focus on +computational aspects and applications. +Holonomic functions are ubiquitous in the sci- +ences. +Examples of holonomic functions include many special functions like error func- +tions, Bessel functions, generalized hypergeometric functions, and linear combinations of +elementary functions. +Plenty of computer algebra systems contain libraries for com- +putations around D-ideals and holonomic functions, such as the Mathematica packages +GeneratingFunctions and HolonomicFunctions [17], the package ore algebra in SAGE, +the built-in DEtools-FindODE in Maple which incorporates HolonomicDE from the package +FPS [16], the package Dmodules.m2 [18] in Macaulay2 [9], and the D-module libraries [1] in +Singular [4], just to name a few. The class of holonomic functions is well-behaved: it is closed +under addition, multiplication, taking integrals and derivatives, and convolution—whenever +defined—and some more operations. However, it is for instance not closed under taking +compositions or reciprocals. In order to decide if the reciprocal of a univariate function is +holonomic, one can make use of the following characterization from [10]. +Proposition 2.2. Let f be holonomic. Its reciprocal 1/f is holonomic iff f ′/f is algebraic. +Example 2.3. Let f = cos . Clearly, f is holonomic of order 2, since f ′′ + f = 0. To that +ODE, one associates the differential operator P = ∂2 + 1 ∈ D. Its reciprocal g = 1/f is +not holonomic, since g has infinitely many poles which cannot appear as singular locus of an +operator in D. By Proposition 2.2, we deduce that f ′/f = − sin/cos = − tan is not algebraic. +However, this further implies that tan itself is not holonomic since all derivatives of tan can +be expressed as polynomials in tan. But one can compute a quadratic differential equation +for g by hand, or by using FPS:-QDE of [34], for instance. One finds the second-order ODE +g(x)g′′(x) − 2(g′(x))2 − (g(x))2 = 0 +(2.5) +of degree 2. +⋄ +Example 2.3 also is an example of a non-holonomic composition f1 ◦ f2 of the holonomic +functions f1(x) = 1/x and f2(x) = cos(x). Those considerations motivate to pass on to +differential equations of higher degree. +2.2 +Differential algebra +Differential algebra studies differential equations that express a polynomial relation between +a function and its derivatives. In this subsection, we recall basic definitions and properties +of differential algebra that will be relevant in the next sections. +4 + +Given a field (or ring) F, a derivation is a map δ: F → F that satisfies δ(f +g) = δ(f)+δ(g) +and Leibniz’ rule δ(f · g) = fδ(g) + δ(f)g for all f, g ∈ F. A differential field (or ring) is +a tuple (F, δ). For a natural number j, we denote by f (j) := δj(f) the j-th derivative of f, +and by f (0) = f. In the multivariate case, one would define a differential polynomial ring +with commuting derivations (see Section 5.2), but we here restrict our presentation to the +univariate case. In this article, we will mainly consider the cases F = K[x] or F = K(x) +together with the derivation δ = ∂ := ∂/∂x. +In differential algebra, the underlying object of study is the differential polynomial ring +F[y(∞) +1 +, y(∞) +2 +, . . . , y(∞) +n +], which corresponds to the set of polynomials in the indeterminates yi +and their derivatives y(j) +i , i = 1, . . . , n, j ∈ N. Below, we give a formal definition. +Definition 2.4. Let (F, δ) be a differential field or ring. The ring of differential polynomials +in y over F, denoted F[y(∞)], is the following differential ring. It is the polynomial ring in +infinitely many variables y, y′, y′′, . . . +F[y(∞)] := F[y, y′, y′′, y(3), . . .] +(2.6) +together with the derivation δ(y(j)) := y(j+1), extending the derivation from F. +In this setting, y is called the differential indeterminate. The ring of differential polynomi- +als in several differential indeterminates y1, . . . , yn is defined by iterating this construction. +The order of a non-zero differential polynomial p ∈ K[y(∞)] is the largest integer n such that +the coefficient of some monomial in p containing y(n) is non-zero. +Definition 2.5. An ideal I ⊂ F[y(∞)] is called differential ideal if p ∈ I implies p′ ∈ I. +For p1, . . . , pk ∈ F[y(∞)], the ideal +⟨p(∞) +1 +, . . . , p(∞) +k +⟩, +(2.7) +where p(∞) +i +denotes the set {p(j) +i }j∈N, is a differential ideal. Moreover, this is the smallest +differential ideal containing p1, . . . , pk, and we will denote it by ⟨p1, . . . , pk⟩(∞). +In the sequel, we will also need truncated version of the differential polynomial ring. For +j ∈ N, we denote by F[y(≤j)] the differential ring +F[y(≤j)] := F[y(∞)]/⟨y(j+1)⟩(∞) ∼= F[y, y′, . . . , y(j)]. +(2.8) +In particular, δ(y(j)) = 0 in F[y(≤j)]. For a differential ideal I = ⟨p1, . . . , pn⟩(∞) ⊂ F[y(∞)] +and j ∈ N, we will denote by I(≤j) the ideal +I(≤j) := ⟨p1, p′ +1, . . . , p(j) +1 , . . . , pn, p′ +n, . . . , p(j) +n ⟩. +(2.9) +Also dynamical models fit well into that setting. Among others, they commonly arise in +chemical reaction networks, see for instance [19] for many examples. Recall that a dynamical +model over F (see [22, Section 1.7], [21, Section 2.2]) is a system of the form +y′ = A(y, u), +z = B(y, u), +(2.10) +5 + +where y = (y1, . . . , yn), z = (z1, . . . , zm), and u = (u1, . . . , ul) are function variables re- +ferred to as the state, output, and input variables, respectively; A ∈ F[y1, . . . , yn, u1, . . . , ul]n, +B ∈ F[y1, . . . , yn, u1, . . . , ul]m. The dimension of system (2.10) is n. The system can be +generalized to the case where A and B are vectors of rational functions. +We here are interested in dynamical models that relate to special D-algebraic functions. +We consider systems M over F = K(x) of the form +y′ = A(y), +z = B(y), +(2.11) +also called state-space system without input, where A = (A1, . . . , An) ∈ K(x)(y1, . . . , yn)n +is a vector of rational functions, B ∈ K(x)(y1, . . . , yn), and y is the vector (y1, . . . , yn). In +order to put (2.11) in the context of differential algebra, let Q be the common denominator +of the system and write Ai = ai/Q for 1 ≤ i ≤ n, and B = b/Q, where a1, . . . , an, b ∈ +K(x)[y1, . . . , yn]. We consider the following n + 1 differential polynomials: +Q y′ − a(y), Q z − b(y) ∈ F[y(∞), z(∞)]. +(2.12) +Before recalling results from [22] in Propositions 2.6 and 2.7, slightly adapted, we recall +that the saturation of an ideal I in a ring R by an element s ∈ R is the ideal +I: s∞ := {r ∈ R | ∃ n ∈ N : sn r ∈ I} . +(2.13) +Geometrically, if R is a polynomial ring, the process of saturation removes those irreducible +components from the algebraic variety defined by I where s vanishes. In our differential +setting, this will avoid that the denominator of fractions of differential polynomials vanishes. +Now let M be a model as in Equation (2.12). +Proposition 2.6. Consider the differential ideal +IM := ⟨Q y′ − a(y), Q z − b(y)⟩(∞): Q∞ ⊂ F[y(∞), z(∞)] . +(2.14) +(1) On the differential polynomial ring K(x)[y(∞), z(∞)], consider the lexicographic monomial +ordering ≺ corresponding to any ordering on the variables such that +(a) z(j1) > yi(j2) for all j1, j2 ∈ N and i ∈ {1, . . . , n}, +(b) z(j+1) > z(j) and y(j+1) +i1 +> y(j) +i2 for all i1, i2, j ∈ N. +Then the set of all the derivatives of (2.12) forms a Gr¨obner basis of IM w.r.t. ≺. +(2) As a commutative algebra, K(x)[y(∞), z(∞)]/IM is isomorphic to K(x)[y1, . . . , yn]. In +particular, IM is a prime differential ideal. +The following proposition will be a key statement to prove that our algorithms terminate. +Proposition 2.7. Consider a non-zero polynomial p in the elimination ideal IM ∩ F[z(∞)] +of lowest degree among the non-zero polynomials of the lowest order in z. Define +IM,j := ⟨(Q y′ − a(y))( ADE1:=diff(y(x),x)^3+y(x)+1=0: +> ADE2:=diff(z(x),x)^2-z(x)-1=0: +> NLDE:-arithmeticDalg([ADE1,ADE2],[y(x),z(x)],w=y+z) +in Maple. +⋄ +Theorem 5.2. Algorithms 1, 2, 3, 4, and 5 are correct. +Proof. The statement follows from combining Proposition 2.7 with Sections 4.1 and 4.2. +Remark 5.3 (Comparison of Methods I and II). We here give some remarks to compare +Methods I and II. In some cases, there are subtle differences between the differential ideals +considered in Methods I and II. The ideal I(≤j) +α +from Method I contains the ideal IMα,j from +Equation (2.15) without the saturation carried out. Moreover, if both p and q are l.h.o., +I(≤j) +α +is contained in the saturated ideal IMα,j. Indeed, the two methods perform differently +in some cases. Consider for instance the two differential polynomials +p = y′′y − y′2 +and +q = y′2 + y2 + 1 . +(5.2) +Our second method finds the differential polynomial +−y y′′ − y y(4) + y′2 + 2 y′ y(3) − y′′2 − y′′ y(4) + y(3)2 +(5.3) +19 + +for α = + in less than a second, whereas the computations based on the first method in +Macaulay2 did not terminate even after a whole day. A Maple implementation of the first +method enabled us to find a third-order differential polynomial of degree 18 in about half a +minute. In our experiments, Method I in general returned ADEs of lower order and higher +degree than Method II in the non-l.h.o. case, and both methods seem to perform similarly +in the l.h.o. case. +⋄ +Remark 5.4 (Polynomial solutions). We now present how one can deal with polynomial +solutions of low degree. Let p = y′′y − y′2 and q = y′2 + y2 + 1, as in Remark 5.3. The only +problematic polynomial zero of p is the zero polynomial, and for q the constant functions +±i, i2 = −1. Then f ± i, where f is a zero of p, is not a solution of (5.3). This can be seen +for f = exp, for instance. We find differential polynomials for these two cases through the +unary operations y ± i over Q(i), and we get +±i y′′ + y y′′ − y′2 . +(5.4) +Alternatively, one can use any of our two methods with the algebraic equation of ±i as an +ADE of order 0 as second input. +⋄ +5.2 +Applications +We here showcase computations for some D-algebraic functions from practical applications, +among others from high energy physics and biology. +Example 5.5 (Elliptic functions). Again, let ℘(x) denote the Weierstrass elliptic function. +It fulfills the ADE +y′(x)2 = 4y(x)3 − c1y(x) − c2, +(5.5) +where c1 and c2 are constants depending on the periods of ℘. Now consider the rational +function κ of the Weierstrass elliptic function +κ(x) = −3a1a13a24℘(x) + a2 +1s1(a2, a3, a4) − 2a1s2(a2, a3, a4) + 3s3(a2, a3, a4) +−3a13a24℘(x) + 3a2 +1 − 2a1s1(a2, a3, a4) + s2(a2, a3, a4) +(5.6) +from the study of Feynman integrals on elliptic curves in [3, Section 7.1]. In (5.6), sn denotes +the elementary symmetric polynomial of degree n in three variables and aij = ai − aj. We +compute an ADE of the following form for κ: +C5 κ′(x)2 = C0 + C1 κ(x)4 + C2 κ(x)3 + C3 κ(x)2 + C4 κ(x) , +(5.7) +where C0, . . . , C5 are polynomials in a1, a2, a3, a4, c1, c2. We hence recover an ADE of the +same form as [3, Equation (7.12)]. +⋄ +Example 5.6 (Epidemiology). The epidemic stage of the SIR (Susceptible-Infected- +Removed) model is given by the system of differential equations: +S′ = −β S I − δ S + µ, +I′ = β S I − γ I + ν, +R′ = δ S + γ I. +(5.8) +20 + +This model describes how a disease can spread within a population. We refer our readers +to [30] for details about the parameters δ, γ, µ, ν. The derivation used is ∂/∂t, where t repre- +sents the time. Suppose we want to compute the minimal differential equation for R. Then +SystoMinDiffPoly will take the triple +[−β S T − δ S + µ, β S T − γ T + ν, δ S + γ T] +(5.9) +as first argument, where I is replaced by T because the variable I is protected for the +imaginary number; as second argument, the function depending on S, T, and R for which +we seek a differential equation; the third argument is [S, T, R] which represents the main +functions of the system whose derivatives are given in the same order in (5.9); and finally, +the dependent variable for the sought differential equation, we choose f(t) since R is already +used among the variables of the system. So the syntax together with the output is: +> timing,p:= CPUTime(NLDE:-SystoMinDiffPoly([-beta*S*T-delta*S + mu, beta*S* +T - gamma*T + nu, delta*S+gamma*T],R,[S,T,R],f(t))):timing +0.485 +> PDEtools:-difforder(p,t) +3 +The output differential equation of our Maple implementation, here p, is of order 3 and +degree 4. +We do not display p as it requires about ten lines. +Note, however, that the +code above computes the differential polynomial in about half a second. Such equations are +important to study certain states of the input system. +⋄ +Example 5.7 (Painlev´e transcendent I). With our second method, we will compute third- +order differential equations for the exponential and the square root of the Painlev´e transcen- +dent of type I. This special function is particularly interesting, since it cannot be expressed +in terms elementary functions or well-known special functions. We will use y′−y and 2xy′−y +as input for exp and √·, respectively. For the exponential, we run +> ADE1:= diff(y(x), x) - y(x) = 0: +> ADE2:= diff(z(x),x,x)=6*z(x)^2+x: #the transcendent +> NLDE:-composeDalg([ADE1,ADE2],[y(x),z(x)],w(x)) +to obtain the ADE +24xw′(x)2w(x)4 + w(x)6 − 2w(x)5 w(3)(x) + 6w′′(x)w′(x)w(x)4 + w(3)(x) +2w(x)4 +−4w′(x)3 − 24w′′(x)w′(x)2w(x)3 − 6w(3)(x)w′′(x)w′(x)w(x)3 + 24w′(x)4w(x)2 ++4w(3)(x)w′(x)3w(x)2 + 9w′′(x)2w′(x)2w(x)2 − 12w′′(x)w′(x)4w(x) + 4w′(x)6 = 0 . +For the square root, we run +> ADE3:= 2*x*diff(y(x), x) - y(x) = 0: +> NLDE:-composeDalg([ADE3,ADE2],[y(x),z(x)],w(x)) +21 + +to obtain the ADE +−48x2w′(x)2w(x)3 + 24xw(x)4 w′(x) − 2xw(3)(x) +2w(x)3 − 4xw′′(x)w(3)(x)w′(x)w(x)2 ++8xw′(x)3w(3)(x)w(x) + 6xw′(x)2w′′(x)2w(x) + 24xw′(x)4w′′(x) + 2w′′(x)w(3)(x)w(x)3 +−3w(x)5 + 2w′(x)2w(3)(x)w(x)2 − 2w′′(x)2w′(x)w(x)2 − 10w′(x)3w′′(x)w(x) −8w′(x)5 = 0 +of order 3 and degree 5. +⋄ +Acknowledgments +We thank Manuel Kauers, Gleb Pogudin, and Bernd Sturmfels for insightful discussions +and Marc H¨ark¨onen for help with our implementation in Macaulay2. 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They are available at https://mathrepo.mis.mpg.de/DAlgebraicFunctions, where +we also provide Jupyter notebooks demonstrating how to use them. +A +Method I in Macaulay2 +-- The following code computes an ADE for the sum of two D-aglebraic functions. +needsPackage "Jets"; +m = ; +n = ; +R = QQ[y_0..y_m, z_0..z_n, w_0, MonomialOrder => Lex]; +P = ; +Q = ; +SumDalg = (P, Q) ->( +T = ideal 0_R; N =0; +I = ideal (P, Q, w_0 - z_0 - y_0); +while +T == ideal 0_R do ( +N = N + 1; +-- Constructing the differential ideal using Jets +J = jets(N, I); +(RingJets,iota) = flattenRing ring(J); +factorialMap = map(RingJets, RingJets, +gens RingJets / ( i-> i => i/(value(substring(1, toString first baseName i)))! )); +-- Substituting the variables with factorials in the coefficients to make the ideal differential +F = factorialMap iota J; +-- Changing the ring from the ring of Jets to a new ring +R = QQ[y_0..y_(N+m), z_0..z_(N+n), w_0..w_N, MonomialOrder => Lex]; +-- Adapting the indices +idxList = gens RingJets/(i->(last baseName i)+value((toString first baseName i)_(-1))); +letterList = gens RingJets / (i -> (toString first baseName i)_0); +mapList = apply(letterList, idxList, (i,j) -> value( i | "_" | toString(j))); +phi = map(R, RingJets, mapList); +L = splice(toList (y_0..y_(N+m), z_0..z_(N+n))); +-- Eliminating the respective variables. +T = eliminate(L, +phi F); +); +return T; +); +--- For other operations, one can use the same code and change the ideal I accordingly. +24 + +--- The next functions are used to construct the polynomials S_i from Equation (4.12). +--- For Monomials +SpecialDifMonomials += ( M ) ->( +if isUnit M == true or M == 0 then +return 0 +else ( +for i from 0 when i < N + 2 do ( +if M % z_i == 0 then (return z_(i + 1) * M // z_i + z_i * SpecialDifMonomials ( M // z_i);); +if M % y_i == 0 then (return y_(i + 1) * z_1 * M // y_i + y_i * SpecialDifMonomials( M // y_i);) +); +); +); +--- For Polynomials +SpecialDifPoly = ( M ) ->( +L = terms (M); +S = {0}; +for i from 0 when i < length L do ( S = append( S, SpecialDifMonomials(L_i))); +return sum S; +); +--- Applying the previous function several times +HigherDif = (M, n) -> ( +if n == 0 then return M; +if n == 1 then ( +return (SpecialDifPoly(M)); +); +return HigherDif(HigherDif (M, 1), n - 1); +); +--- The next code computes an ADE for the composition of two D-algebraic functions. +needsPackage "Jets"; +m = ; +n = ; +R = QQ[y_0..y_m, z_0..z_n, x_0..x_1, MonomialOrder => Lex]; +P = ; +Q = ; +CompDAlg = (P,Q) -> ( +I = ideal(P, Q, x_1 -1 ); +T = ideal 0_R; N =0; +while T == ideal 0_R do ( +N = N + 1; +-- Constructing the differential ideal using Jets +J = jets(N, I); (RingJets,iota) = flattenRing ring(J); +factorialMap = map(RingJets, RingJets, +gens RingJets / ( i-> i => i / (value(substring(1, toString first baseName i)))! )); +-- Substituting the variables with factorials in the coefficients to make the ideal differential +F = factorialMap iota J; +-- Changing the ring from the ring of Jets to a new ring +R = QQ[w_0..w_(N + min(m,n)), y_0..y_(N+m), z_0..z_(N+n), x_0..x_(N+1), MonomialOrder => Lex]; +-- Adapting the indices +idxList = gens RingJets/(i->(last baseName i)+value((toString first baseName i)_(-1))); +letterList = gens RingJets / (i -> (toString first baseName i)_0); +mapList = apply(letterList, idxList, (i,j) -> value( i | "_" | toString(j))); +phi = map(R, RingJets, mapList); +H = phi F; -- Constructing the ideal containing the differential equation for the composition +for i from 0 when i < N + 2 do ( +H = H + ideal( w_i - HigherDif(y_0, i) ); +); +L = splice(toList (y_0..y_(N+m), z_0..z_(N+n), x_1..x_(N +1) )); +T = eliminate(L, H); -- Eliminating the respective variables +); +return T; +); +25 + +B +Method II in Maple +############################################################################################## +## The following is the source code of the Maple package NLDE which exports four commands: +## +1. arithmeticDalg: for arithmetic operations, i.e., addition, multiplication, division, +## +exponentiation with D-algebraic functions. In general, for rational +## +functions of finitely many D-algebraic functions. +## +2. unaryDalg: +for rational functions of a single D-algebraic functions +## +3. ComposeDalg: +for compositions of two D-algebraic functions +## +4. SystoMinDiffPoly: for finding input-output equations of a dynamical system +############################################################################################## +NLDE:= module() +option ‘Copyright (c) 2022 the-third-author, Max Planck Institute for MiS, Leipzig‘, package; +export unaryDalg, SystoMinDiffPoly, composeDalg, arithmeticDalg; +local buildsystem, mergesystem, ftogh, subsgfurther; +buildsystem:= proc(DE::‘=‘,y::anyfunc(name),x::name,$)::list(‘=‘); +local t::name, d::posint, SubL::list, PolDE::polynom, j::nonnegint; +option ‘Copyright (c) 2022 the-third-author‘; +t:=op(y); +d:=PDEtools:-difforder(DE,t); +#variables of substitution for the model, the input x with indices +SubL:=[seq(diff(y,[t$j])=x[j],j=0..d)]; +PolDE:=subs(SubL,lhs(DE)); +#the differential equation is not LEF +if degree(PolDE,x[d])>1 then +d:=d+1; +SubL:=[op(SubL),diff(y,t$d)=x[d]]; +PolDE:=subs(SubL,lhs(diff(DE,t))); +return [[seq(x[j],j=1..(d-1)),solve(PolDE,x[d])],[seq(x[j],j=0..(d-1))]] +else +return [[seq(x[j],j=1..(d-1)),solve(PolDE,x[d])],[seq(x[j],j=0..(d-1))]] +end if +end proc: +mergesystem:= proc(L::list(‘=‘),V::list(anyfunc(name)),$)::‘=‘; +local l::posint:=numelems(L), j::posint, Sys::list, vars::list, deriv::list, +n::posint, x::nothing, X::list, i::posint, Ind::list; +option ‘Copyright (c) 2022 the-third-author‘; +Sys:=[seq(buildsystem(L[j],V[j],cat(x,j)),j=1..l)]; +vars:=map(r->op(r[2]),Sys); +deriv:=map(r->op(r[1]),Sys); +n:=numelems(vars); +X:=[seq(vars[j]=x[j],j=1..n)]; +#indices of the variables representing the solutions of the input DEs +Ind:=[seq(1+add(numelems(Sys[i][2]),i=1..(j-1)),j=1..l)]; +return [subs(X,deriv),map(r->x[r],Ind),map(rhs,X)] +end proc: +SystoMinDiffPoly:= proc(f::list(algebraic),g::algebraic,X::Or(list,set), +z::anyfunc(name),{ordering::identical(plex,lexdeg):=plex},$)::algebraic; +local F,G,q1,q2,Q,Svars,J1,J2,J,n,Xt,t,DE,Sub:=[],allvars,yvars,ord,j,k,y,alpha; +option ‘Copyright (c) 2022 the-third-author‘; +t:=op(1,z); +y:=op(0,z); +alpha:=indets([f,g]) minus {op(X)}; +n:=numelems(X); +F:=normal(f); +q1:=mul(map(denom,F)); +G:=normal(g); +26 + +q2:=denom(G); +#least common multiple of the denominators of the system +Q:=lcm(q1,q2); +#to differentiate, the variables should be functions of the +#independent variable t +Xt:=map(x->x=x(t),X); +Q:=subs(Xt,Q); +F:=subs(Xt,F); +G:=subs(Xt,G); +Xt:=map(rhs,Xt); +J1:=[seq(Q*diff(Xt[j],t)-normal(Q*F[j]),j=1..n)]; +#differentiating n-1 times the polynomials Q*x’-Q*f +for j to n do: +J1:=[op(J1),seq(diff(J1[j],t$k),k=1..(n-1))] +end do; +#differentiating n times the polynomials Q*y - Q*g +J2:=[seq(diff(Q*y(t)-normal(Q*G),[t$j]),j=0..n)]; +J:=[op(J1),op(J2)]; +#build the list of substitution to see derivatives as variables +for j to n do: +Sub:=[op(Sub),seq(diff(Xt[j],[t$k])=x[j,k],k=0..n)] +end do; +if ordering=plex then +#elimination and saturation with Groebner bases +#w.r.t. pure lex monomial ordering +Sub:=[op(Sub),seq(diff(y(t),[t$j])=y[j],j=0..n)]; +allvars:=ListTools:-Reverse(map(rhs,Sub)); +yvars:=select(has,allvars,y); +allvars:=allvars[numelems(yvars)+1..-1]; +J:=PolynomialIdeals:-PolynomialIdeal(subs(Sub,J),parameters=alpha); +J:=PolynomialIdeals:-Saturate(J,subs(Sub,Q)); +J:=Groebner:-Basis(J,plex(op(allvars),op(yvars))); +J:=remove(has,J,allvars) +else +#elimination and saturation with Groebner bases +#w.r.t. lexdeg elimination ordering +Sub:={op(Sub),seq(diff(y(t),[t$j])=y[j],j=0..n)}; +J:=PolynomialIdeals:-PolynomialIdeal(subs(Sub,J),parameters=alpha); +J:=PolynomialIdeals:-Saturate(J,subs(Sub,Q)); +yvars:=select(has,map(rhs,Sub),y); +J:=PolynomialIdeals:-EliminationIdeal(J,yvars); +J:=select(type,convert(J,list),polynom) +end if; +#Taking a diff polynomial of minimal total degree +# among those of the minimal order +J:=map(de->collect(de,[seq(y[j],j=0..n)],’distributed’),J); +Sub:=select(has,map(e->rhs(e)=lhs(e),Sub),y); +J:=map(de->subs(Sub,de),J); +#order +ord:=min(map(de->PDEtools:-difforder(de,t),J)); +DE:=select(de->PDEtools:-difforder(de,t)=ord,J); +Sub:=map(e->rhs(e)=lhs(e),Sub); +DE:=map(de->subs(Sub,de),DE); +#degree +DE:=sort(DE,(a,b)->degree(a,yvars)<=degree(b,yvars)); +DE:=DE[1]; +Sub:=map(e->rhs(e)=lhs(e),Sub); +return subs(Sub,DE)=0 +end proc: +subsgfurther :=proc(gm1::algebraic,g::name,t::name,m::posint,n::posint,$)::list; +local k::posint,j::nonnegint,Subdiff::list,rSubdiff::list,eqg,Sub::list; +option ‘Copyright (c) 2022 the-third-author‘; +k:=0; +eqg:=gm1; +#the first substition +27 + +Sub:=[g[m]=eqg]; +if krhs(r)=lhs(r),Subdiff)),diff(g(t),t$m)=g[m]]; +#the other substitutions are obtained by +#differentiating and substituting the first substitution +while ksubs([seq(diff(g(x),[x$j])=g[j],j=0..N)],r),Lderiv); +Lderiv:= map(r->subs(g[0]=x,r),Lderiv); +Lderiv:= map(r->convert(r,diff),Lderiv); +Lderiv:= map(r->subs([seq(diff(f(x),[x$j])=f[j],j=0..N)],r),Lderiv); +return SolveTools:-Linear(Lderiv,[seq(f[j],j=0..N)]) +end proc: +composeDalg:= proc(L::[‘=‘,‘=‘],V::[anyfunc(name),anyfunc(name)],z::anyfunc(name), +{ordering::identical(plex,lexdeg):=plex},$)::‘=‘; +local t::name:=op(z),n::posint,fgh::set(‘=‘),R,f::nothing,Sys::list,x::nothing, +Sysh::list,g::nothing,h::nothing,j::nonnegint,m::posint,Subg::list, +DE1::‘=‘,DE2::‘=‘; +option ‘Copyright (c) 2022 the-third-author‘; +DE1:=lhs(L[1])-rhs(L[1]); +DE2:=lhs(L[2])-rhs(L[2])=0; +n:=PDEtools:-difforder(DE1,t); +#substitution to remove derivatives and t from the first differential equation (DE) +R:=subs(t=g[0],subs([seq(diff(V[1],[t$j])=f[j],j=0..n)],DE1)); +#in case on non-LEF, one derivation is needed +if degree(R,f[n])>1 then +n:=n+1; +R:=subs(t=g[0],subs([seq(diff(V[1],[t$j])=f[j],j=0..n)],diff(DE1,t))) +end if; +#build the system with the second DE +Sys:=buildsystem(DE2,V[2],g); +m:=numelems(Sys[1]); +#expressing derivatives of f in terms of those of h and g +fgh:=ftogh(f,g,h,x,n); +if m<=n then +#build a list of further substitutions in R if m<=n +Subg:=subsgfurther(Sys[1][m],g,t,m,n); +fgh:=subs(Subg,fgh) +end if; +#the full substitution (f is already eliminated here) +R:=normal(subs(fgh,R)); +#the system for h is easily obtained from R +Sysh:=[[seq(h[j],j=1..(n-1)),solve(R,h[n])],[seq(h[j],j=0..(n-1))]]; +#use SystoMinDiffPoly to return the desired output +return SystoMinDiffPoly([op(Sys[1]),op(Sysh[1])],h[0],[op(Sys[2]),op(Sysh[2])],z,’:-ordering’=ordering) +end proc: +28 + +unaryDalg:= proc(DE::‘=‘,y::anyfunc(name),z::name=ratpoly, +{ordering::identical(plex,lexdeg):=plex},$)::‘=‘; +local t::name:=op(y),var::name:=op(0,y),dvar::name=lhs(z), +r::ratpoly:=rhs(z),eq::algebraic,j::nonnegint,Sys::list,x::nothing; +option ‘Copyright (c) 2022 the-third-author‘; +var:=op(0,y); +dvar:=lhs(z); +r:=normal(rhs(z)); +#Simple case: y appears linearly in r(y) +if degree(numer(r),var)<=1 and degree(denom(r),var)<=1 then +eq:=subs(dvar=dvar(t),solve(dvar-r,var)); +eq:=eval(lhs(DE) - rhs(DE), y=eq); +eq:=numer(normal(eq)); +return collect(eq,[seq(diff(dvar(t),[t$j]), +j=0..PDEtools:-difforder(DE,t))],’distributed’)=0 +end if; +#General case +#build the system using buildsystem +Sys:=buildsystem(lhs(DE) - rhs(DE)=0,y,x); +#use SystoMinDiffPoly to return the desired output +return SystoMinDiffPoly(Sys[1],subs(var=Sys[2][1],r),Sys[2],dvar(t),’:-ordering’=ordering) +end proc: +arithmeticDalg:=proc(L::list(‘=‘),V::list(anyfunc(name)),z::name=ratpoly, +{ordering::identical(plex,lexdeg):=plex},$)::‘=‘; +local t:=op(1,V[1]),DEs::list(‘=‘),Sys::list,j::posint,subV::list; +option ‘Copyright (c) 2022 the-third-author‘; +if numelems(L)=1 then +return L +end if; +DEs:=map(r->lhs(r) - rhs(r)=0,L); +#build the systems and merge them using mergesystem +Sys:=mergesystem(DEs,V); +#prepare the list for the change of variables +#in r according to Sys +subV:=[seq(op(0,V[j])=Sys[2][j],j=1..numelems(V))]; +#use SystoMinDiffPoly to return the desired output +return SystoMinDiffPoly(Sys[1],subs(subV,rhs(z)),Sys[3],lhs(z)(t),’:-ordering’=ordering) +end proc: +end module: +savelib(’NLDE’,"path_to_your_folder_for_softwarepackages/NLDE.mla"): +Authors’ addresses: +Rida Ait El Manssour, MPI MiS♭ +rida.manssour@mis.mpg.de +Anna-Laura Sattelberger, MPI MiS♭ and KTH♯ (current) +alsat@kth.se +Bertrand Teguia Tabuguia, MPI MiS♭ +bertrand.teguia@mis.mpg.de +♭ Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany +♯ Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden +29 + diff --git a/a9E0T4oBgHgl3EQfngEk/content/tmp_files/load_file.txt b/a9E0T4oBgHgl3EQfngEk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b7930ade5a5f6b88e625907cf9ef1b14dd472896 --- /dev/null +++ b/a9E0T4oBgHgl3EQfngEk/content/tmp_files/load_file.txt @@ -0,0 +1,1687 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf,len=1686 +page_content='D-Algebraic Functions Rida Ait El Manssour, Anna-Laura Sattelberger, Bertrand Teguia Tabuguia Abstract Differentially-algebraic (D-algebraic) functions are solutions of polynomial equa- tions in the function, its derivatives, and the independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' We revisit closure properties of these functions by providing constructive proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' We present algorithms to compute algebraic differential equations for compositions and arithmetic manipula- tions of univariate D-algebraic functions and derive bounds for the order of the resulting differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' We apply our methods to examples in the sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' 1 Introduction The Weyl algebra encodes linear differential operators with polynomial coefficients, such as the operator P = ∂2 − x arising from Airy’s differential equation f ′′(x) − xf(x) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='1) Solutions of such differential equations are called holonomic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Differential algebra investigates polynomials in a differential indeterminate y and its derivatives with coefficients in a differential ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Those polynomials are called differential polynomials and their as- sociated differential equations are commonly referred to as algebraic differential equations (ADEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' For instance, the differential polynomial p = y′2 − 4y3 − c1y − c2, where c1, c2 are constants, encodes the differential equation (y′(x))2 = 4(y(x))3 + c1y(x) + c2 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='2) of which the Weierstrass elliptic function ℘ (see [24, 13]) is a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' ADEs also nat- urally arise in structural identifiability, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Functions which are zeros of differential polynomials—or, equivalently, solutions of the corresponding ADEs—are called D-algebraic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' This class of functions arises in a natural way from the study of holonomic func- tions: for instance, the reciprocal of a holonomic function is in general not holonomic, but it is D-algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' The present article can hence be located at the interim of the theory of D-ideals and differential algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' For introductions to those fields, we refer our readers to [29, 28] and [25, 14], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Although they are present at several places in the literature (see [26, 31, 27, 15]), it is difficult to find exhaustive expositions about D-algebraic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' This could be explained by the challenging computational complexity observable from [11], which is detrimental for applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' One attempt of computer algebra is to find subclasses 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='02512v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='AG] 6 Jan 2023 of D-algebraic functions that offer both a mathematical structure and efficient algorithms for the corresponding arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Some relevant subclasses are holonomic functions, which are also called “D-finite functions” (see [17, 32]), DD-finite functions that satisfy differential equations with D-finite coefficients [12], or functions that satisfy ADEs of degree at most 2 [34, 33], called “δ2-finite functions” therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' While the rich theory of D-modules and holo- nomic functions covers the degree-one case, similar constructions for ADEs of higher degree are still in an early stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' The idea of bounding the degree of the ADEs is to alleviate the complexity compared to the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' The class of D-algebraic functions has nice closure properties, which is for instance carried out in [23] and [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' The former illustrates the use of D-algebraic functions in combinatorics for studying generating functions, with an emphasis on quadrant walks via Tutte’s invariant method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' The latter focuses on the zero test prob- lem for D-algebraic functions viewed as formal power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' The earliest appearance of the terminology “differentially algebraic” that we are aware of is in Rubel’s work [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' In this article, we construct ADEs for compositions and rational functions of D-algebraic functions, taking only the ADEs of the original functions as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' We give bounds for the order of the resulting differential polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Finding an ADE fulfilled by a rational expression in a D-algebraic function f, or finding an ADE fulfilled by its antiderivative, are unary operations in the input ADE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' The rest of operations—like sums, products, ratios, and compositions—are binary operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' However, we implemented them for arbitrarily many operands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Our algorithms take differential polynomials p, q as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' From those, we construct ADEs which rational functions, antiderivatives, and compositions of all solutions of the input ADEs fulfill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' We stress that working with the differential polynomials implies working with the set of all solutions of the corresponding ADEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Gr¨obner basis theories exist both for D-ideals and for differential ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Viewing ADEs as differential polynomials, one can encode an operation between D-algebraic functions by an ideal in a differential polynomial ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Such an ideal is differentially generated by the given polynomials and a rational expression built from the underlying operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' We use jets to truncate the obtained differential ideal at desired order before applying elimination theory based on Gr¨obner basis techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' We compare it to a second method that still proceeds with Gr¨obner bases, but uses bounds for the order of the resulting differential polynomial, which we establish in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' The main contribution of this article is the development of two strategies to compute differential equations satisfied by rational expressions and com- positions of D-algebraic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' In particular, our algorithms and their implementations are general, reliable, and outperform existing algorithms and software, which often do not contain such computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' For instance, the find_ioequations command of the Julia [2] package StructuralIdentifiability can be used to derive ADEs with constant coeffi- cients only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' moreover, that command requires a user-defined dynamical model, which is not needed in our algorithms for arithmetic operations and compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' For each operation with D-algebraic functions, we provide a bound for the order of the resulting differential polynomial in Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='9 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='12 which depends on those of the input ADEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' We imple- mented our first method in Macaulay2 [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' The second one is implemented as the Maple [20] package NLDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Our implementations are made available via the MathRepo [6]—a repository website hosted by MPI MiS—at https://mathrepo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='mis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='de/DAlgebraicFunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' 2 Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Our article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Section 2 recalls basic concepts about the Weyl algebra as well as differential algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' In Section 3, we present D-algebraic functions and investigate their closure properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' In Section 4, we study the arithmetic of those functions: we construct ADEs for antiderivatives, compositions and rational expressions of D-algebraic functions and study the order of the resulting differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Section 5 presents pseudocode to carry those operations out in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' We present applications of our results in the study of Feynman integrals and epidemiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' The code for our implemen- tations in Macaulay2 and Maple is provided in Appendices A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' By N, we denote the nonnegative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Letters f, g, h are reserved for functions, D denotes the Weyl algebra, P ∈ D is used for linear differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Dif- ferential polynomials are denoted by lower case letters p, q, r, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' They are polynomials in dependent variables u, y, z, w and the independent variable x (or x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , xk in the multivari- ate case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' “Degree” means the total degree of polynomials, unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' By a rational expression in f, we will mean a rational function in x, f, and the derivatives of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' 2 Algebraic aspects of differential equations We briefly recall some algebraic aspects of differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' We here revisit two classes of differential equations, namely linear and algebraic differential equations with polynomial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Throughout this section, we describe the case of univariate functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='1 The Weyl algebra and holonomic functions The Weyl algebra, denoted D := K[x]⟨∂⟩, is the free K-algebra generated by x and ∂ modulo the following relation: the commutator of ∂ and x is [∂, x] = ∂x − x∂ = 1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='1) which encodes Leibniz’ rule for taking the derivative of a product of functions in a formal way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Hence D = � k � i=0 ai∂i | k ∈ N, ai ∈ K[x] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='2) The order of a differential operator P = � i ai∂i is ord(P) := max {i | ai ̸= 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='3) For P ∈ D, the vanishing locus of the leading polynomial aord(P) is the singular locus of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' To a differential operator P, one associates the ODE P(f) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=', one looks for functions f that are annihilated by the differential operator P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' A function f(x) is holonomic if there exists a differential operator P ∈ D which annihilates f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' 3 Holonomicity of D-modules dates back to Bernstein and Kashiwara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Effective compu- tations with holonomic functions were first studied by Zeilberger [36] for proving identities between special functions automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' The name “D-finite” is justified by the following observation: a function f is holonomic if and only if the K(x)-vector space spanned by the derivatives of f is finite-dimensional, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=', if dimK(x) � spanK(x) �� ∂k(f) � k∈N �� < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='4) In general, a (uni- or multivariate) function is called holonomic if its annihilating D-ideal is holonomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' We refer to [28, 29] for an introduction to D-modules with a focus on computational aspects and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Holonomic functions are ubiquitous in the sci- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Examples of holonomic functions include many special functions like error func- tions, Bessel functions, generalized hypergeometric functions, and linear combinations of elementary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Plenty of computer algebra systems contain libraries for com- putations around D-ideals and holonomic functions, such as the Mathematica packages GeneratingFunctions and HolonomicFunctions [17], the package ore algebra in SAGE, the built-in DEtools-FindODE in Maple which incorporates HolonomicDE from the package FPS [16], the package Dmodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='m2 [18] in Macaulay2 [9], and the D-module libraries [1] in Singular [4], just to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' The class of holonomic functions is well-behaved: it is closed under addition, multiplication, taking integrals and derivatives, and convolution—whenever defined—and some more operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' However, it is for instance not closed under taking compositions or reciprocals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' In order to decide if the reciprocal of a univariate function is holonomic, one can make use of the following characterization from [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Let f be holonomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Its reciprocal 1/f is holonomic iff f ′/f is algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Let f = cos .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Clearly, f is holonomic of order 2, since f ′′ + f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' To that ODE, one associates the differential operator P = ∂2 + 1 ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Its reciprocal g = 1/f is not holonomic, since g has infinitely many poles which cannot appear as singular locus of an operator in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='2, we deduce that f ′/f = − sin/cos = − tan is not algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' However, this further implies that tan itself is not holonomic since all derivatives of tan can be expressed as polynomials in tan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' But one can compute a quadratic differential equation for g by hand, or by using FPS:-QDE of [34], for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' One finds the second-order ODE g(x)g′′(x) − 2(g′(x))2 − (g(x))2 = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='5) of degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' ⋄ Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='3 also is an example of a non-holonomic composition f1 ◦ f2 of the holonomic functions f1(x) = 1/x and f2(x) = cos(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Those considerations motivate to pass on to differential equations of higher degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='2 Differential algebra Differential algebra studies differential equations that express a polynomial relation between a function and its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' In this subsection, we recall basic definitions and properties of differential algebra that will be relevant in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' 4 Given a field (or ring) F, a derivation is a map δ: F → F that satisfies δ(f +g) = δ(f)+δ(g) and Leibniz’ rule δ(f · g) = fδ(g) + δ(f)g for all f, g ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' A differential field (or ring) is a tuple (F, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' For a natural number j, we denote by f (j) := δj(f) the j-th derivative of f, and by f (0) = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' In the multivariate case, one would define a differential polynomial ring with commuting derivations (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='2), but we here restrict our presentation to the univariate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' In this article, we will mainly consider the cases F = K[x] or F = K(x) together with the derivation δ = ∂ := ∂/∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' In differential algebra, the underlying object of study is the differential polynomial ring F[y(∞) 1 , y(∞) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , y(∞) n ], which corresponds to the set of polynomials in the indeterminates yi and their derivatives y(j) i , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , n, j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Below, we give a formal definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Let (F, δ) be a differential field or ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' The ring of differential polynomials in y over F, denoted F[y(∞)], is the following differential ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' It is the polynomial ring in infinitely many variables y, y′, y′′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' F[y(∞)] := F[y, y′, y′′, y(3), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='6) together with the derivation δ(y(j)) := y(j+1), extending the derivation from F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' In this setting, y is called the differential indeterminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' The ring of differential polynomi- als in several differential indeterminates y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , yn is defined by iterating this construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' The order of a non-zero differential polynomial p ∈ K[y(∞)] is the largest integer n such that the coefficient of some monomial in p containing y(n) is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' An ideal I ⊂ F[y(∞)] is called differential ideal if p ∈ I implies p′ ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' For p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , pk ∈ F[y(∞)], the ideal ⟨p(∞) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , p(∞) k ⟩, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='7) where p(∞) i denotes the set {p(j) i }j∈N, is a differential ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Moreover, this is the smallest differential ideal containing p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , pk, and we will denote it by ⟨p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , pk⟩(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' In the sequel, we will also need truncated version of the differential polynomial ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' For j ∈ N, we denote by F[y(≤j)] the differential ring F[y(≤j)] := F[y(∞)]/⟨y(j+1)⟩(∞) ∼= F[y, y′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , y(j)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='8) In particular, δ(y(j)) = 0 in F[y(≤j)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' For a differential ideal I = ⟨p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , pn⟩(∞) ⊂ F[y(∞)] and j ∈ N, we will denote by I(≤j) the ideal I(≤j) := ⟨p1, p′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , p(j) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , pn, p′ n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , p(j) n ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='9) Also dynamical models fit well into that setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Among others, they commonly arise in chemical reaction networks, see for instance [19] for many examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Recall that a dynamical model over F (see [22, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='7], [21, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='2]) is a system of the form y′ = A(y, u), z = B(y, u), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='10) 5 where y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , yn), z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , zm), and u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , ul) are function variables re- ferred to as the state, output, and input variables, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' A ∈ F[y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , yn, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , ul]n, B ∈ F[y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , yn, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , ul]m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' The dimension of system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='10) is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' The system can be generalized to the case where A and B are vectors of rational functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' We here are interested in dynamical models that relate to special D-algebraic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' We consider systems M over F = K(x) of the form y′ = A(y), z = B(y), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='11) also called state-space system without input, where A = (A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , An) ∈ K(x)(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , yn)n is a vector of rational functions, B ∈ K(x)(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , yn), and y is the vector (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , yn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' In order to put (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='11) in the context of differential algebra, let Q be the common denominator of the system and write Ai = ai/Q for 1 ≤ i ≤ n, and B = b/Q, where a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , an, b ∈ K(x)[y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , yn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' We consider the following n + 1 differential polynomials: Q y′ − a(y), Q z − b(y) ∈ F[y(∞), z(∞)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='12) Before recalling results from [22] in Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='7, slightly adapted, we recall that the saturation of an ideal I in a ring R by an element s ∈ R is the ideal I: s∞ := {r ∈ R | ∃ n ∈ N : sn r ∈ I} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='13) Geometrically, if R is a polynomial ring, the process of saturation removes those irreducible components from the algebraic variety defined by I where s vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' In our differential setting, this will avoid that the denominator of fractions of differential polynomials vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Now let M be a model as in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Consider the differential ideal IM := ⟨Q y′ − a(y), Q z − b(y)⟩(∞): Q∞ ⊂ F[y(∞), z(∞)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='14) (1) On the differential polynomial ring K(x)[y(∞), z(∞)], consider the lexicographic monomial ordering ≺ corresponding to any ordering on the variables such that (a) z(j1) > yi(j2) for all j1, j2 ∈ N and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , n}, (b) z(j+1) > z(j) and y(j+1) i1 > y(j) i2 for all i1, i2, j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Then the set of all the derivatives of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='12) forms a Gr¨obner basis of IM w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' ≺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' (2) As a commutative algebra, K(x)[y(∞), z(∞)]/IM is isomorphic to K(x)[y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' , yn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' In particular, IM is a prime differential ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' The following proposition will be a key statement to prove that our algorithms terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Consider a non-zero polynomial p in the elimination ideal IM ∩ F[z(∞)] of lowest degree among the non-zero polynomials of the lowest order in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E0T4oBgHgl3EQfngEk/content/2301.02512v1.pdf'} +page_content=' Define IM,j := ⟨(Q y′ − a(y))(